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УДК 512.552.1
DOI 10.24147/1812-3996.2021.26(2).12-22
ОБ ОДНОЙ МОДИФИКАЦИИ ПОНЯТИЙ ПОЛНОТЫ, РЕДУЦИРОВАННОСТИ, ПЕРИОДИЧНОСТИ И ПРИМАРНОСТИ ДЛЯ АССОЦИАТИВНЫХ КОЛЕЦ
Л. М. Мартынов
Омский государственный педагогический университет, г. Омск, Россия
Аннотация. Введенные автором в 1996 г. для произвольных алгебр понятия полноты, редуцированности, периодичности и примарности модифицируются для ассоциативных колец так, что модифицированные периодические ассоциативные кольца являются в точности кольца с конечными однопорожденными подкольцами и все конечные поля являются модифицированными примарными кольцами. Изучаются свойства этих модифицированных кольцевых понятий. В частности, характеризуются модифицированные периодические, примарные и редуцированные многообразия ассоциативных колец.
Ключевые слова
Ассоциативное кольцо, полное кольцо, редуцированное кольцо, периодическое кольцо, примарное кольцо
ABOUT ONE MODIFICATION OF CONCEPTS OF COMPLETELY, REDUCIBILITY, PERIODICITY AND PRIMARITY FOR ASSOCIATIVE RINGS
L. M. Martynov
Omsk State Pedagogical University, Omsk, Russia
Abstract. Introduced by the author in 1996 for arbitrary algebras, the concepts of completeness, reducibility, periodicity and primarity are modified for associative rings so that the modified periodic associative rings are exactly rings with finite one generated subrings and all finite fields are modified primary rings. Properties of these modified ring concepts are studied. In particular, modified periodic, primary, and reduced varieties of associative rings are characterized.
Available online 24.09.2021
Keywords
Associative ring, complete ring, reduced ring, periodic ring, primary ring
Информация о статье
Дата поступления 10.03.2021
Дата принятия в печать 13.07.2021
Дата онлайн-размещения 24.09.2021
Article info
Received 10.03.2021
Accepted 13.07.2021
The theory of Abelian groups gives a vivid example of the advanced structural theory. An important role in it is played by the concepts of a complete (divisible), reduced, periodic (in particular, primary) group. It turns out there is alternative approach to these notions which uses the theory of group varieties. This allowed us in [1] (also see [2] and [3]) to define analogs of the mentioned concepts for arbitrary algebras. In these works, concepts of periodic and primary algebra are based on concepts of complete and reduced algebras. In the case of groups or semigroups, the concept of periodic algebra
turns in the generally accepted concept of a periodic group or semigroup as an algebra with finite monogenic (i.e., one generated) subalgebras. We would like to have a similar situation at least in the case of associative rings. In our approach, when we started from the set At(L(As)) of atoms of the lattice of subvarie-
ties of the variety Ms of all associative ring, any nonsimple finite field, being a monogenic associative ring, was not periodic. In [3], we drew attention to the fact that sometimes it can be a useful modification of the concepts under discussion, when instead of the set of
atoms of the lattice of subvarieties of a given variety of algebras, another set M of its subvarieties is considered.
In this article, we modify the concepts of completely, reducibility, periodicity and primarity by considering a special set M of subvarieties of the variety Ms, which is obtained by adding to the set At(L(Ms)) all varieties, each of which is generated by some finite nonsimple field. In this case, ^-periodic rings are rings with finite monogenic subrings (here there is an analogy with groups and semigroups) and any finite field is ^-periodic and even ^-primary. Properties of ^-periodic and ^-primary associative rings are studied. In addition, we characterize the ^-periodic, ^-primary and ^-reduced varieties of associative rings. The last turn out finitely ^-reduced and ^-periodic.
Note that the concepts of (atomic) primarity and (atomic) reducibility were studied in the author's work [4] for monoassociative algebras. Let's also pay attention to the fact that in the theory of associative rings, a periodic ring is often understood as a ring whose multiplicative semigroup is periodic (in the usual sense) (see, for example, [5]).
1. The Main Definitions and Notations
We will agree in the future that a ring means an associative ring and an ideal means a two-side ideal. Let
V be a variety of rings. A ring R is called V-complete if the V-verbal ideal V(R) (i.e., the smallest ideal in the set of all, factor rings of R for which belongs V) coincides with R. A ring is called V-solvable if it has no nonzero V-complete subrings.
The notion of V-solvable ring R can be defined using the V-verbal series of R [6; 7]. Namely, for every ordinal a we define the a-th V-verbal subring Va(R) of R by transfinite induction: V0(fl) = R; if a is not a limit ordinal then Va(R) = V(Va-1(R)); and if a > 0 is a limit ordinal then Va(R) = ^p<aVP(R). We thus obtain a decreasing chain of subrings of R:
R = V0(R) > V1(R) > ••• > Va(R) > ■■■, which is called to as the V-verbal chain of R. Clearly, Va+1(R) is an ideal in Va(R), but the V-verbal chain of R is not a chain of its ideals in general.
This chain stabilizes for some ordinal y, i.e., VY(R) = VY+1(R). In this case we say that the variety
V is y-attainable on R. If there exists an ordinal y such that V is y-attainable on all rings then V is called y-at-tainable (in Ms) [6; 7]. The term "attainability" (in our terminology, this is 1-attainability) was first introduced by T. Tamura [8] and considered by A. I. Maltsev [9].
If VY(R) = {0} for some y then the ring R is called y-V-solvable and the least y with this property is called
the step of V-solvability of R. Clearly, the ring R is V-solvable if and only if R is y-V-solvable for some ordinal y. If y is a natural number then y-V-solvable ring is also called finitely V-solvable.
For a set M by IMI we agree to denote the cardinal number of M. The letters k, I, m, n (sometimes with indices) denote (if there are no reservations) positive integers and the letter p denotes a prime number.
We agree that R0 is denoted the ring with the zero multiplication which is obtained from the ring R by replacing its multiplication with zero multiplication. By R+ is denoted the additive group of the ring R. If M is a subset of the ring R, then (M) and (M) denote the subring and the ideal of R generated by M, respectively. If a £ R and M = {a}, then the subring generated by M is called monogenic and denoted by (a). An element e £ R with the property e2 = e is called an idempotent of R.
By 1(X) we agree to denote a free ring in M.s over the infinite countable set = {x1, x1,...}, i.e., the ring of polynomials with integer coefficients in non-commuting variables of X with zero free terms. The identity is understood as a formal equality of the form f(x1,x2,...,xn) = 0 where f(x1,x2,...,xn)£ Z(X). Let var£ denote the variety of rings defined by a system £ of ring identities and varK be the least variety of rings containing a class K of rings (in other words, varK is the variety generated of K). The various variables x1 and x2 will often be denoted by the letters x and y, respectively. In particular, a monogenic ring free in Ms will be denoted by 1(x).
Below we use the following notations:
P is the set of prime numbers;
H+ is the set of all positive integers;
H0 is the set of all non-negative integers;
V is a variety of rings;
L(V) is the lattice of subvarieties of V;
At(L(V)) is the set of atoms of the lattice L(V);
Fpm (= GF(pm )) is a finite (Galois) field of order
pm;
Tpm = var{px = 0,xp = x} = varFpm;
Hn is the ring of residues modulo n;
Zn = var{nx = 0,xy = 0} = varZ^.
Let M be the union of two sets T and Z varieties of rings where T= {Tpm | p £ P, m £ Z+}, Z=[Zp\ p £ P}, i.e., M = TUZ. Note that the set Mcontains the set At(L(Ms)) which consists, as is well known [10], of varieties of the form Tp and Zp for all prime p.
We will say that a ring R is M-primary (relative to M £ M) or M-primary ring if each of its monogenic subring is finitely M-solvable. We say that a ring R is
M-primary if R is M-primary for some M £ M. It is obvious that a ring of any variety M £ M is M-primary.
A ring R is called M-complete if R is M-complete for every M £ M. We call a ring M-reduced if R has no non-trivial ^-complete subrings. By analogy, the concepts of ^-complete [^-complete] and ^-reduced [^-reduced] are defined
Recall that if the variety V is given by the identity system £, then the verbal V(R) of the ring R coincides with the ideal of R generated by all possible values in R of all polynomials the left-hand sides of the
identities of £. For varieties Z,
Tpm and a ring R,
we indicate formulas to calculate the coresponding verbals: Zp(R) = pR + R2, ypm(R) = pR + + (ap — a | a £ R). It is clear that the ring R is ^-complete if and only if Zp(R) = R and Tpm(R) = R for any p and m.
The fact that a ring A is a subring [ideal] of a ring R is denoted either by A^R [A < fl] or by R> A [fl > A]. A subring [ideal] A of R is called proper and denoted either by A < R [A < fl] or by R > A [R o A] if A ± R.
Recall that the product (in sense of Maltsev [9]) of classes X and y of rings is a class £° y of rings defined as follows:
R £X° y O (31 <R)(I £X & R/I£y).
It is known [9] that if X and y are varieties of rings, then the class X ° y is a variety. If X ° X = X, we say that X is closed with respect to extensions or X is an idempotent of the groupoid (L(<A&), °). As usual, we will say that ring R is an extension of a ring A by a ring B if there exists such ideal I of R that I = A and A/I = B.
A finite decreasing sequence subring of a ring R:
R = I0!>I1 !>■■■!> Ik!> Ik+1 > ••• > In = {0} (1.1)
we will call subideal series of R, if Ik+1 is an ideal in Ik for any k = 0,1,2, ...,n — 1. If all the members of the series (1.1) are ideals of the ring R, then it is called an ideal series of R.
We say that a ring R is finitely M-reduced if for some not necessarily distinct varieties M1t M2, ..., Mn from M the ring R possesses a finite decreasing subideal series whose factors Ik/Ik+1 (k = 0,1,2, ...,n— 1) belong to Mk+1 (see also [11]). In this case we will also say that the ring R is finitely M-reduced by varieties M1t M2, ..., Mn from the set M.
We remind that a ring R is called transverbal [9] by a variety V of rings if for any ideal I of R its V-verbal V(I) is also an ideal of R. If any ring is transverbal by V,
then the variety <A& will be call transverbal by the sub-variety V. It is easy to see that the variety Ms is transversal by any variety from the set M. In fact, this is obvious for varieties of the form Zp for any p: if I < R, then Zp(I) =pl + I2 < R. For varieties of the form Tpn, this follows from well-known Andrunakievich's Lemma [12, Lemma 4] according to which, from I < R, and J < I, and the factor ring /// does not have non-zero nilpotent elements, it follows that J < R. It easily follows that finitely reduced by varieties Mv M2, ..., Mn from the set M the ring R has an ideal series R=T0!>T1 !>■■■!> Tk > Tk+1 > • > Tn = {0} (1.2) obtained by analogy with [13] from the series (1.1) by Tk+1 = Mk (Tk) with the factors Tk/Tk+1 from Mk (k = 0,1,2,.,.,n—1).
It is appropriate to note one important fact: any finitely M-reduced ring R is reduced. Indeed, the ring R has an ideal series of the form (1.2) with factors belonging to varieties M1t M2, ..., Mn from M. If we assume that the ring R is not ^-reduced, then it contains a nonzero ^-complete subring C of R. For any variety M of M there is M(C) = C. On the other hand, for any ring R and its subring C, the relationship M(C) ^ M(R) is obvious, from which, in our case, it is follows the relationship C ^ M(R). Now it is clear that any member Tk (k = 0,1,2,..., n) of the ideal series (1.2) will contain subring C of R and this contradicts the fact that Tn = {0}.
We say that a ring R is M-periodic if any monogenic subring of R is finitely M-reduced. We point out the connection of this concept with the product of varieties from M. For any sequence V2, ..., Vn,... of not necessarily distinct varieties of rings, we define the product N4=^1 = V1°V2° ■■■ °Vn as follows: nhVi=V1 and Uf=1Vi =(№1^) °Vn when n = 2,3,... It is easy to understand (see also [11]) that a ring R finitely M-reduced by varieties M1t M2, ...,Mn from M if and only if R belongs to the product of these varieties in reverse order, i.e., R £ Mn ° Mn-1 ° ■•• ° M2 °M1. In particular, this means that the class of all rings, each of which is M-reduced by varieties M-v M2, ...,Mn from M, is a variety. Without going into details, in such cases we will talk about varieties of finitely ^-reduced rings of the same type.
Notice that if a ring R is M-primary, then any monogenic subring of R is finitely M-solvable and therefore finitely ^-reduced by a finite sequence
M . M ,.... M for some к. Thus, for any of M of Ж, any
к times
M-primary ring is M-periodic.
Following [14], we call a variety V of rings is pre-complete if the lattice L(V) of subvarieties of V has a single atom, If this atom is T then we call V precomplete relative to T or P-precomplete.
Following [15], we will say that a nilpotent ring R has an index of nilpotency n if n is the smallest natural number such that Rn = O. The index of nilpotency of a variety V of rings is the upper face of indices of nilpotency of its nilpotent rings and ^ if there is no such number.
Me will say that a nil ring R has a nil index n if n is the smallest natural number such that an = 0 for any a E R. The nil index of a variety V of rings is the upper face of nil indices of its nil rings and ^ if there is no such number.
Below we use the following notations:
Z = var[xy = 0} is the variety of rings with the zero multiplication;
M is a variety from the set Ж;
T is an atom of the lattice L(As);
charR is a characteristic of R, i.e., the smallest n El+ with the property nR = О or the number 0 if there is no such n;
CharV is a characteristic of V, i.e., a characteristic of the V-free monogenic ring F1(V);
Жкп = var{kx = 0,xn = 0} is the variety of nil rings of characteristic к and nil indexes n;
is the variety of solvable rings of step < n and characteristic k;
Note that all varieties of rings with the zero multiplication form the sublattice L(Z) in which is isomorphic to the lattice Ь(Л)) of varieties of Abelian groups. We agree to call Z-solvable rings simply solvable.
A ring is called a Ml-ring if each of its non-nilpotent monogenic subrings contains a non-zero idempotent. Clearly, every Ml-ring is an l-ring in the sense of [16, p. 304], i.e., a ring in which each non-nil right ideal contains non-zero idempotent. Every nil ring is obviously an Ml-ring. We call a ring an essential Ml-ring if it is an Ml-ring which is not a nil ring.
The intersection of all non-zero ideals of a ring is its monolith. If the monolith of a ring is a non-zero ideal then the ring is called subdirectly indecomposable.
The symbols t and I represent the beginning and the end of the proof of the corresponding statement, respectively. The symbol I is also used when the proof is omitted. If a variety V consists of rings that possess some property expressed by an adjective then we add
this adjective to the name of V. For example, the phrase "V is a Ж-primary variety" means that V consists of rings each of which is M-primary relative to some fixed variety M in Ж.
2. General remarks
Here we note properties of the concepts of Ж-completeness, Ж-reducibility and finite Ж-reduci-bility for rings that are necessary for further presentation of the material. Some of them are similar to those noted at the end of the article [2] for the corresponding concepts of atomic completeness and reducibility and proved there for arbitrary algebras.
Lemma 2.1. For any variety V of rings, a homo-morphic image of a V-complete ring is a V-complete ring. In particular, a homomorphic image of a Ж-com-plete ring is a Ж-complete ring.
t Indeed, if С is a V-complete ring and I is an ideal of C, then V(C/I) = (V(c) + I) /I = (C +I)/I = = C/I. I
Lemma 2.2. For any variety V of rings, the extension of a V-complete ring by a V-complete is a complete ring. In particular, the extension of a Ж-complete ring by a Ж-complete is a Ж-complete ring.
t Indeed, let R be a ring, I <R, V(I) = I, and V(R/I ) = R/I. Then (R/I ) = (V(R) + I)/I = = R/I. It follows that V(R) + I = R. On other hand, I <R implies I = V(I) < V(R) and therefore V(R) + + I = V(R) = R. I
Lemma 2.3. The direct sum of any set of Ж-complete rings is a Ж-complete ring.
t Let R =®iEi Ri where Ri is Ж-complete ring for any i E I. It is easy to see that the equality M(R) =®iE, M(Rt) holds for any variety M E Ж. Since the equality M(Rt) = Rt for any i E I is true by the condition, we obtain from the preceding equality that M(R) = R, and this means the completeness of the ring R. I
The following statement follows from the definition of a reduced ring.
Lemma 2.4. A subring of a of Ж-reduced is a Ж-reduced ring. I
Lemma 2.5. The direct product of any set of Ж-reduced rings is a Ж-reduced ring.
t Let R = П;Е/ Ri where Rt is Ж-reduced ring for any i E I. If we assume that the ring R has a non-zero Ж-complete subring, then for at least one index i E I the ring R had a non-zero projection on Rt, which by Lemma 2.1 would be a non-zero Ж-complete subring of the ring Rt, which contradicts its Ж-reducibility. Thus, R is a Ж-reduced ring. I
Lemma 2.6. The extension of a M-reduced ring by a M-reduced ring is a M-reduced ring.
t Let I be such ^-reduced ideal of a ring R that the quotient ring R/I is ^-reduced and C be a ^-complete subring of R. Then by Lemma 2.1, the natural ho-momorphism of R onto R/I maps C of R into the zero ring so R/I is ^-reduced. Therefore, C £ /. This fact and ^-reducibility of I imply that C is the zero ring. Thus, R is a ^-reduced ring. I
We note a few more properties of the concept of finite ^-reducibility. The first of these was noted and proved in Section 1.
Lemma 2.7. Any finitely M-reduced ring R is M-reduced. I
Lemma 2.8. The direct sum of any finite set of finitely M-reduced rings is a finitely M-reduced ring.
t Let R =®1i=1 Rt where Rt is a finitely ^-reduced ring by varieties Mi1, Mi2, ..., Mik. from M for any i £ 1, m. It is easy to see that the ring R is a finitely ^-reduced ring by varieties M11, M12,
..., M1kl,M21, M22.....M2k2.....Mm1, Mm2.....Mmkm
from M and this means that the ring R is finitely ^-reduced. I
The following lemma is obvious. Lemma 2.9. If the quotient-ring A/I of a ring A by an ideal I is finitely M-reduced by varieties from a set £ M and the ideal I is finitely M-reduced by varieties from a set M2 £ M, then the ring A is finitely M-reduced by varieties from the set (Mt U M2) £ №. In other words, an extension of a finitely M-reduced ring by a finitely M-reduced ring is a finitely M-reduced ring. I
It is use to have in view the following two obvious statements.
Lemma 2.10. A ring R with zero multiplication is M-complete if and only if R is Z-complete, equivalently, the additive group R+ is complete (divisible). I
Lemma 2.11. A ring with zero multiplication is M-reduced if and only if R is Z-reduced, equivalently, the additive group R+ is reduced. I
Lemma 2.12. If the additive group R+ of the ring R is not reduced, then R is not M-reduced.
t In the case of non-reducibility of the additive group fl+of the ring R, R+ contains a non-zero subgroup A with the property pA = A for any p. Hence, it is easy to conclude that for the non-zero subring (,4> of the ring R and any variety M of M, takes place A £ M((A>) and therefore M((A>) = (A>, i.e., <4> is the ^-complete ring. I
Lemma 2.13. Any ring R with zero multiplication of a non-zero characteristics is finitely Z-reduced.
Let charR = k > 1 and k = pf1p22 ■■■Pml is canonical decomposition of a natural number k into the product of prime numbers. The additive Abelian group R+ of the ring R is the finite direct sum of the cyclic primary groups of orders p^1, p22,—, p^ respectively, i.e., R+= C(p^1)SiC(pl22)m.^C(pl^m). But then R = C^p*1)0® C(p22)°® ... © C(p^)°is the
direct sum of rings C(pi l) (i £ 1,m) with zero multiplication. Given that for any zero-multiplication ring A, prime p, and variety Zp from Z holds Zp(A) = pA + A2 = pA, it is easy to understand that the ring R is finitely reduced by a sequence
ZP1, ZP1, .■ ■, ZP1, ZP2, ZP2, .■ ■, ZP2, .■■,ZPm, ZPm, .■ ■ , ZPm
k1 times
k2 times
km times
of varieties from Z. I
Lemma 2.14. Any nilpotent ring R of a non-zero characteristic is finitely Z-reduced.
t Let charR = k > 1 and Rn = 0 for some n> 1. Consider a Z-verbal series of the ring R: R = R(0) > R(1) > ■•• > R(i) > R(i+1) > ■•• > R(m) = = {0}, (2.1)
where R(i+1) = (R(i))2(i= 0,1,...,n — 1) and m > | The factors R(i)/R(i+1) for any i £ 0,m — 1) are rings with zero multiplication of a non-zero characteristics and therefore, by Lemma 2.12, they are finitely Z-re-duced. Compacting Z-solvable ideal series (2.1) of the ring R with preimages under natural homomorphism R(l) onto R(l)/R(l+1) of the corresponding ideal series of rings R(i)/R(i+1) for any £ 0,m — 1) , we obtain the ideal series with factors from the varieties belonging to Z. Thus, the ring R is finitely Z-reduced. I
Lemma 2.15. Any nil ring C coincident with its square is M-complete.
t Since the homomorphic image of a nil ring is a nil ring, C is ^-complete. On the other hand, for any p, we have Zp(C) = pC + C2 = C and hence C is Z-com-plete. Thus C is ^-complete. I
We conclude this section with the following two obvious statements.
Lemma 2.16. If a simple ring R doesn't belong to a variety V of rings, then R is V-complete. In particular, if a simple ring R does not belong to every variety M from M, then R is M-complete. I
For example, the ring of all square matrices of order n > 1 over any field is ^-complete. Any infinite field is also such.
Corollary 2.17. A simple ring R is M-reduced if and only if R isomorphic to Fpn or Zp for some p, n. I
3. Periodic Rings and Varieties
Recall that a ring is called locally finite if any finitely generated subring of its is finite. A variety of rings is called locally finite if it consists of locally finite rings.
The following statement is true.
Proposition 3.1. A ring is Ж-periodic if and only if any its monogenic subring is finite.
Before proving Proposition 2.1, for convenience reading, we formulate several statements, some of which is known, and their consequences for our case.
The first lemma was noted in [17, Corollary 2.1] for rings, but it is true for any algebraic systems [18, Theorem 4, p. 361].
Lemma 3.2. A ring variety generated of a finite ring is locally finite. I
Corollary 3.3. The set Ж consists of locally finite ring varieties.
t Indeed, any variety of Ж is generated by a finite ring. I
Lemma 3.4. ([17, Proposition 3.1]) A product of locally finite varieties of rings is a locally finite variety of rings. I
Lemma 3.5. Any finitely Ж-reduced ring is locally
finite.
t Indeed, if R is a Ж-reduced by varieties M^ M2, ..., Mn from Ж ring , as it notes in Section 1,
R belongs to the product Mn ° Mn
•• °M1 which,
due to Corollary 3.3 and Lemma 3.4, is a locally finite variety. Therefore R is the locally finite ring. I
Corollary 3.6. Any finitely Ж-reduced monogenic ring is finite. I
Lemma 3.7. Any finite commutative ring R is finitely Ж-reduced.
t Indeed, the Jacobson radical J(R) of the ring R is nilpotent [16, Theorem 1, p. 63] and by Lemma 2.14 ](R) is finitely ^-reduced. On the other hand, the finite semisimple commutative quotient ring R/](R) is direct sum of finite fields [16, Theorem 2, p. 32] and therefore it is finitely ^-reduced. By Lemma 2.9, the ring R is finitely Ж-reduced. I
Proof of Proposition 2.1.
t ^ If a ring R is Ж-periodic, then any monogenic subring A of R is finitely Ж-reduced and therefore by Corollary 3.6 A is finite.
^ Conversely, let any monogenic subring A of a ring R is finite. Since A is a commutative ring, by Lemma 3.7 A is a finitely Ж-reduced ring. This means that R is periodic ring. I
The main result of this section is the following Theorem 3.8. For varieties of associative rings V the following conditions are equivalent:
(1) V is a periodic variety;
(2) V is a locally finite variety;
(3) all monogenic ring of V are finite;
(4) V satisfies the identities
xk = xl (k > I ), nx = 0. (2.1)
t The equivalence of conditions (2) - (4) is proved in [17, Proposition 2.1]. We prove the equivalence of the conditions (1) and (3).
(1) ^ (3) Let V be a periodic variety. Then by Proposition 2.1 any monogenic ring A of V is finite.
(3) ^ (1) Let all monogenic rings of V are finite and R be any ring of V. Then any monogenic subring A of R is finite commutative ring. By Lemma 3.7 A is a finitely ^-reduced ring. This means that R is periodic ring. Therefore, V consists of periodic rings. I
It is appropriate to note here two important consequences from Lemma 2.13 and Proposition 3.1.
Corollary 3.9. Any nil ring R of a non-zero characteristic is Z-periodic and therefore for any k, n, the variety n consists from Z-periodic rings.
t Indeed, for any a £ R, monogenic subring A = (a) of R is nilpotent of non-zero characteristic and therefore, by Lemma 2.14, A is finitely ^-reduced. By Proposition 3.1, A is finite I
Corollary 3.10. Any solvable ring R of a non-zero characteristic is Z-periodic and hence for any k,n, the variety consists from Z-periodic rings.
t For the proof, it is sufficient to note that R is nilpotent, and therefore R is a nil ring of non-zero characteristic, and to refer to Corollary 3.9. I 4. Primary Rings and Varieties First of all, recall that as noted in Section 1, that any ^-primary ring is periodic in particular, any monogenic subring of such ring is finite. The following statement is true. Proposition 4.1. A ring R is M-primary relative to the variety M £ M (i. e., R is a M-primary ring) if and only if one of the conditions is hold:
(1) M = Tpm for some p, m, and R £ Tpm;
(2) M = Zp for some p and for any a £ R there is k,n such that pka = 0 and an = 0.
t ^ Let R be a M-primary ring. Consider the possible cases.
If M = Tpm for some p, m, then by Lemma 10 from [6], Tpm o fpm = Tpm. This fact means that any monogenic subring of the ring R, being finitely Tpm-solvable, belongs to a variety Tpm. In particular, R satisfies the identities px = 0 and xp™ = x and therefore R £ Tpm. Thus, in this case, the condition (1) is satisfied.
о
Let M = Zp for some , a £ R, and A = (a). Since a finitely Zp-solvable ring is obviously nilpotent, there is such n that an = 0. On the other hand, it is also obvious that the additive group of the finite ring A must be a primary p-group and therefore there exists such k that pka = 0. Thus, in this case, the condition (2) is hold.
^ Conversely, if M = Fpm for some p, m, and R £ Fpm then any monogenic subring of the ring R belongs to Fpm and therefore the ring R is a Fpm-primary ring.
If M = Zp for some p and for any a £ R there is k,n such that pka = 0 and an = 0, then monogenic subring A = (a) of the ring R is commutative nilpotent and from the proofs of Lemmas 2.13 and 2.14 it follows easily that R is finitely 2Tp-reduced. The latter is equivalent to finite Zp-solvability of A, i.e., in this case R is a Zp-primary ring.
So, in both cases, R is a ^-primary ring, I
The main result of this section is the following
Theorem 4.2. A variety V of rings is M-primary if and only if one of the following conditions:
(1) V £ Fpm for some p, m ;
(2) V £ Npkn for some p, k, n.
Before proving Theorem 4.2, we state several lemmas. For convenience reading, we paraphrase Theorem 4 from [14] for rings proved in [14] for associative algebras over a non-zero commutative ring with unit as the following statement.
Lemma 4.3. A variety V of rings is precomplete if only if CharV = pk for some p and k. I
Corollary 4.4. Every variety of the set M is pre-complete.
t Indeed, if M £ M, then CharM = p for some
p. I
Lemma 4.5. If V is a M-primary variety of rings and M contains an atom P, then V is T-precomplete.
t We consider possible cases.
If M £ Z, then T =M = Zp for some p. Suppose that V, in addition to T, contains an atom T' other than P, and let be a free monogenic ring in T'. Then the simple ring F1 doesn't belong to variety T and therefore by Lemma 2.16, F1 is ^-complete. On the other hand, F1 must be finitely ^-solvable. The last is possible only in the case when F1 = 0, that is a contradiction. Thus V is ^-precomplete.
If M £ F, then M = Fpm for some p, n and therefore T = Fp. It is clear that the variety V cannot contain atoms of the form Zp for any p and Fp, for any prime p' ± p, since rings Zp and Fp, are Fpm-complete by Lemma 2.16. I 18 -
Proof of Theorem 4.2.
t ^ We first assume that the variety V is ^-primary relative to M £ T. Then M =Fpm for some p and m. By Proposition 4.1 (1), any ring R of V belong to the variety Fpm. Therefore, in this case V £ Fpm and hence the condition (1) is hold.
Now, assume that V is an ^-primary variety for some M £ Z. Then M = Zp for some p. By Lemma 4.4
V is Zp-precomplete, and by Lemma 4.3 CharV = pk for some k. Then free in V monogenic ring F^V) = (x) has the characteristics pk, i.e., pkx = 0, and F^V) is primary Zp-ring. By Proposition 4.1 (2), F^V) is a nil ring, i.e., xn = 0 for some n Thus in the variety V is hold identities pkx = 0 ,xn = 0 and therefore
V £ Npkn for some p, k, n. Thus, in this case, the condition (2) is hold.
^ Conversely, if the inclusion V £ Tpm occurs for some , m, then according to Proposition 2.1 (1), any ring of V is a Tpm-primary ring. This means that in this case
V is a Tpm-primary variety.
If the inclusion V£Npkn holds for some for some p,k,m, then in the variety V is hold identities pkx = 0 ,xn = 0. According to Proposition 4.1 (2), any ring of V is Zp-primary. This means that in this case, V is a Zp-primary variety.
Thus, in both cases V is a ^-primary variety. I
5. Reduced Varieties of Rings
Let TQ be the set of all subvarieties TQ of Ms each of which is generated by some finite set of finite fields. It is known (see, for example, [20, Theorem 14]) and easily verified that the product X °y of two varieties X,y £ TQ coincides with their union Xvy in the lattice L(Ms) and is also generated by a finite set of finite fields. Thus, the set (TQ, °) is a commutative semigroup of idempotents, i.e., a semilattice. It is also obvious that the set T is the set of generators of this semi-lattice. We also note that, as is well known (see, for example, [6]), varieties with an identity xm = x for some m > 1 are exhausted by varieties from the set TQ. It is obvious that any such variety is ^-reduced.
The main result of this section is the following
Theorem 5.1. A variety V of rings is M-reduced if and only if
V£ S£oTg, (5.1)
for some k, n and TFQ £ TQ.
Before proving Theorem 5.1, we state several auxiliary statements.
Lemma 5.2. Any nil ring R of a №-reduce variety
V is solvable. I
t Indeed, if we assume that R is not solvable, then R contents non-zero nil subring A coincident with its square. Since the ring A belongs to the variety V and A is ^-complete by Lemma 2.15, we get a contradiction with the fact that V is a ^-reduced variety. Thus, the ring R must be solvable. I
The following lemma can be derived from Proposition 1.2 of [19] adapted to our case and from Lemma 2.15.
Lemma 5.3. If a variety V of rings contains a solvable ring of any finite step, then V has a non-zero nil ring coincident with its square and therefore the variety V is not №-reduce. I
Lemma 5.4. The steps of solvable rings in every M-reduced variety V of rings are bounded collectively by a certain natural number. In particular, V has a finite index of nilpotency and a finite nil index. I
t If V contains a solvable ring R of infinite step, then in V there exist solvable ring R/R(n)of any finite step n. By Lemma 5.3 the variety V is not ^-reduce which contradicts the condition of Lemma 5.4. Thus, steps of the ^-solvability of rings in V are uniformly bounded by a natural number n. In particular, V has a finite index of nilpotency. Since the monogenic subrings of finitely solvable rings are finitely solvable and therefore are nil rings, we conclude that the nil indices of the elements of solvable rings from V are bounded collectively by the number k. I
Lemma 5.5. The rings L and I0 do not belong to any M-reduced variety V of rings.
t In fact, if we assume that V contains the ring L0, then V contains the variety varL0 and hence for any prime number p, the ring C^ obtained from the additive quasi-cyclic p-group Cp™ by defining zero multipli- R = Re El R(1-e), where R(1 — e) = {a
Corollary 5.8. Nil rings of a M-reduced variety V of rings form a subvariety of the variety for some k and n. The converse is also true, a subvariety of the variety is finitely Z-reduced. I
The proofs of the following two statements are similar to the proofs of Lemma 13 and Corollary 5 of [4], respectively.
Lemma 5.9. For every M-reduced variety V of rings, there is a number m such that, for every n > m, in the algebra L(x) there is a polynomial f(x) such that the identity xm = xnf(x) is valid in V. I
Corollary 5.10. Every M-reduced variety V of rings consists of Ml-rings. I
The proofs of the following two statements are similar to the proofs of Lemmas 15 and 16 from [4], respectively.
Lemma 5.11. In every M-reduced variety V of rings, the class K of all rings in V without nilpotent elements is a subvariety of the variety var{xm = x} for some m > 1. The converse is also true, for any m > 1, a subvariety of the variety var{xm = x} is finitely T-reduced I
Lemma 5.12. The variety TFQ of rings which is generated by a finite set F of finite fields is finitely T-reduced. I
Lemma 5.13. If the monolith M of a ring R is a field, then R = M.
t Let monolith M of a ring R be a field and e be a unit of M. We show that M lies in the center of R. For any elements a £ R and c £ M, we have ac = a(ec) = = (ae)c = c(ae) = (ca)e = ca.
Consider the Peirce decomposition of R into the direct sum of two additive subgroups:
cation on it. But in view of Lemma 2.10, the ring Cpx is ^-complete. The latter contradicts the ^-reducibility of the variety V. Therefore, a ^-reduced variety V cannot contain a ring L0.
On the other hand, the ring L cannot belong to V, otherwise, for any p and n, the solvable ring (p)/(pn) would belong to V contradicting Lemma 5.4. I
The proof of the following statement is contained in [17, Lemma 2.11] (in a more general situation, it was noted also in [4, Lemma 14]).
Lemma 5.6. If V is a variety of rings, and CharV = 0, then L0 £V. I
Directly from Lemmas 5.6 and 5.5 follows Corollary 5.7. Any M-reduced variety V of rings has a non-zero characteristic. I
Besides, from Lemmas 5.2 and 5.4, Corollary 5.7 and Lemma 2.14 follows
— ae I a £ R} as usual. In our case these subgroups annihilate each other, since for any elements a,x £ R , we have
ae(x — xe) = a(e(x — xe)) = a(ex — xe) = 0.
From here we can easily get that Re and R(1 — e) is an ideal of R. This means that R is the direct sum of its ideals Re and R(1 — e), i.e., R = Re ® R(1 — e). Hence, R(1 — e) = 0 and so R = Re = M. I
Proof of Theorem 5.1.
t ^ Let V be a non-zero ^-reduced variety of rings. By Corollary 5.7, V has a non-zero characteristic k > 1. Note also that by Lemma 5.11, all rings without nilpotent elements satisfy the identity xm = x for some m > 1 and are therefore the subdirect product of finite fields. It is clear that, up to isomorphism, the variety V contains only a finite number of finite fields, and the
prime characteristics of these fields must be divisors of k. Let T(V) be the set of all finite fields of V and TQ = varT(V).
Let R is a ring of V. If R is a nil ring then by Corollary 5.8 R belong to the variety for some n and, in consequence, to the variety o TQ. Suppose that R is not a nil ring. Then by Corollary 5.10, R is an essential MI-ring. Let E be the set of all non-zero idempotents of R. For every idempotent e £ E, let Te denote the maximal ideal of R among all ideals not containing e, and let N be the intersection of all these ideals Te (e £ E). It is clear that N is a nil ideal, coinciding with the Jacobson radical J(R), and the ring A/N is a subdirect sum of the ring Re = R/Te (e £ E). Since every ideal of the ring Re contains the non-zero idempotent e = e + N, the monolith Me of Re contains e and therefore M| is non-zero ideal of Re. Hence, M| = Me. This means that the ring Me is Z-complete. However, V is a ^-reduced variety and therefore must exist a variety T of T for which Ie = T(Me) & Me. Since the ring Me/Ie belongs to the variety T and so has no nilpotent elements, in view of the already mentioned Andrunakievich's Lemma, Ie is an ideal of Re and hence Ie = 0. Thus, the monolith Me £ T. Any non-zero ideal I of a ring Me is an ideal of a ring Re and therefore coincides with Me, since the factor ring Me/I belongs to T and therefore has no nilpotent elements. Thus, Me is a simple commutative ^-reduced ring from T and hence by Corollary 2.17 Me is a finite field of T. By Lemma 5.13, Re = Me and hence Re£T.
This means that R/N is a subdirect product of finite fields from T(V), and so R/N £ TQ = = varT(V)) . On the other hand, by Corollary 5.8 the nil ideal N belongs to the variety for some n. Thus, R £ S£°TQ. By the arbitrariness of the ring R £ V, we conclude that inclusion (5.1) holds.
^ Conversely, let V be a variety of rings and V C ° TQ for some k,n and TQ £ TQ. Then by Corollary 5.8, rings of the varieties is finitely ^-reduced and by Lemmas 5.12, TQ are finitely ^-reduced. Besides, by Lemma 2.6, the variety o TQ is finitely ^-reduced. But then the variety V as subvariety finitely of the ^-reduced variety o TQ is finitely ^-reduced. I
We note an important corollary of Theorem 5.1.
Corollary 5.14. For a variety V of rings, the following conditions are equivalent:
1) V is an M-reduced variety;
2) V is a finitely M-reduced variety.
t 1) ^ 2) If V is an ^-reduced variety then by Theorem 5.1 V C o TQ and, as noted at the end of 20 -
the proof of Theorem 5.1, V is a finitely ^-reduced variety.
2) ^ 1) If V is a finitely ^-reduced variety then by Lemma 2.7, V is ^-reduced. I
Corollary 5.15. Any M-reduced variety of rings is M-periodic.
t If V is a ^-reduced variety then by Corollary 5.14 V is a finitely ^-reduced. Hence, all monogenic rings of V are a finitely ^-reduced and therefore V consists of ^-periodic rings. I
Remark 5.16. The converse to Corollary 5.15 is not true, so, for example, in the periodic variety Mk = var{xn = 0} of nil rings of nil index n>1 there is a non-zero nil ring coinciding with its square, which by the Lemma 2.15 is ^-complete and therefore the variety Mk is not ^-reduced.
Remark 5.17. Note that, according to Lemmas 2.1 and 2.2, the class K of all ^-complete rings is closed with respect to homomorphic images and extensions, and, according to Lemmas 2.4 - 2.6, the class S of all M- reduced rings is closed under subalgebras, direct products, and extensions. It is easy to see that the variety Ms is transversal by any variety from the set M. Having somewhat modified the main result of [22], replacing the set At(L(As)) with the set M, we easily get that the map v: —> Ms, which corresponds to each ring R the largest ^-complete ideal r(R) (= CM(R)), satisfies the conditions axiomatic definition of a radical (in sense Kurosh and Amitsur) (see, for example, [23, p. 91] or [24, p. 27]). We will call CM(R) ^-complete radical of the ring fi. Note, that CM(R) contains any ^-complete subring of R and therefore it is strict radical [24, p. 148]. At the same time, Ms = KoS, i.e., any ring is an extension of ^-complete ring by ^-re-duced ring.
Remark 5.18. The description of the idempotents of the groupoid (Ms, o) given in [20, Theorem 5] was obtained earlier in [6, Lemma 10, Theorem 4]. Theorem 15 of [20], which states that there are no nontrivial ring varieties attainable in ^i^, is incorrect. It contradicts both Theorem 4 of [6], which states that ring varieties attainable in are exhausted by varieties from TQ, and Theorem 5 of [20]. In fact, if we assume that a certain variety T £ TQ is unattainable in Ms, then there is a ring R such that I1 = T(R) & I2 = T(T(R)). Since the quotient ring I1/I2 does not contain non-zero nilpotent elements, then according to the earlier mentioned Andrunakievich's Lemma, I2 is an ideal of R. But then the factor-ring R/I2 belongs to the product T oT (I1/I2 £ T and (R/hVOi/h) = R/h £ T) but R/I2 does not belong to T. The latter contradicts the fact that
according to Theorem 5 of [20] F is idempotent in the groupoid (Ms, o).
Remark 5.19. Concepts of completely, reducibil-ity, periodicity and primarity can be modified for monoassociative algebras over arbitrary associative and commutative ring R with 1^0. In this case, as the set M we must take the union of the set T of all varieties of fl-algebras, each from which generated by a finite
fl-field R/P, where P is a maximal ideal in R of finite index and the set Z of all varieties of fl-algebras of the type ZP = var{Px = 0,xy = 0} = varR° (here Rp = = (R/P)0 for any maximal ideal P). In particular, in this case, for fl-algebras over an infinite field R, M periodic algebras are exactly nil algebras, which is quite natural.
The definitions of the concepts studied here and some results are announced in [25].
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Вестник Омского университета 2021. Т. 26, № 2. С. 12-22
-ISSN 1812-3996
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ИНФОРМАЦИЯ ОБ АВТОРЕ
Мартынов Леонид Матвеевич - доктор физико-математических наук, профессор, профессор кафедры математики и методики обучения математике, Омский государственный педагогический университет, 644099, Россия, г. Омск, наб. Тухачевского, 14; е-таМ: mart@omsk.edu.
INFORMATION ABOUT THE AUTHOR
Martynov Leonid Matveevich - Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of Mathematics and Methods of Teaching Mathematics, Omsk State Pedagogical University, 14, nab. Tukhachevskogo, Omsk, 644099, Russia; e-mail: mart@omsk.edu.
ДЛЯ ЦИТИРОВАНИЯ
Мартынов Л. М. Об одной модификации понятий полноты, редуцированности, периодичности и примарности для ассоциативных колец // Вестн. Ом. унта. 2021. Т. 26, № 2. С. 12-22. Э01: 10.24147/1812-3996.2021.26(2).12-22. (На англ. яз.).
FOR GTATIONS
Martynov L.M. About one modification of concepts of completely, reducibility, periodicity and primarity for associative rings. Vestnik Omskogo universiteta = Herald of Omsk University, 2021, vol. 26, no. 2, pp. 12-22. DOI: 10.24147/1812-3996.2021.26(2).12-22.
