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0URAL MATHEMATICAL JOURNAL, Vol. 10, No. 1, 2024, pp. 84-98
DOI: 10.15826/umj.2024.1.008
ARTINIAN M-COMPLETE, M-REDUCED, AND MINIMALLY M-COMPLETE ASSOCIATIVE RINGS
Leonid M. Martynov
Omsk State University of Ways of Communication, 35 Marksa Ppr., Omsk, 644046, Russian Federation
Tatiana V. Pavlova
University of Tyumen, 6 Volodarskogo Str., Tyumen, 625003, Russian Federation
t.v.pavlova@utmn.ru
Abstract: In 1996, the first author defined analogs of the concepts of complete (divisible), reduced, and periodic abelian groups, well-known in the theory of abelian groups, for arbitrary varieties of algebras. In 2021, the first author proposed a modification of the concepts of completeness and reducibility, which is more natural in the case of associative rings. The paper studies the modification of these concepts for associative rings. Artinian M-complete, M-reduced rings, and minimally M-complete associative nilpotent rings, simple rings with unity, and finite rings are characterized.
Keywords: Associative ring, Artinian ring, Finite ring, Complete ring, Reduced ring.
1. Introduction
In the theory of abelian groups, the notions of complete (divisible), reduced, and periodic (in particular, primary) groups are of great importance. In [16] (see also [17, 19]), some analogs of these notions were defined for arbitrary varieties of algebras. In the mentioned papers, the concepts of (atomic) complete, (atomic) reduced, and solvable algebra [30] (see also [31]) were defined by means of the atoms of these varieties and the Malcev products for these atoms [15]. Furthermore, the notions of periodic and primary algebra were defined using notions of (atomic) completeness, (atomic) reducibility, and solvability. In particular, an algebra is called periodic if each of its monogenic (i.e., one generated) subalgebras is finitely reduced. Note that a group or semigroup is periodic as a universal algebra (in our sense) if and only if it is periodic as a group or semigroup in the ordinary sense.
It is different for associative rings. In the theory of associative rings, a ring is called periodic if its multiplicative semigroup is periodic (in the ordinary sense). Any finite nonprime field is a periodic ring in the ordinary sense. On the other hand, such a field is monogenic, but it is not a finitely reduced ring; i.e., it is not a periodic algebra in the sense of papers [16, 17, 19]. To remove this difference, paper [21] suggests modifying the concepts of complete, reducible, periodic, and primary associative rings. This is done by using a special set M of subvarieties of the variety As of associative rings, where M is the union of the set of lattice atoms of subvarieties of As and the set of all varieties, each of which is generated by some finite nonprime field. In this case, M-periodic rings are rings with finite monogenic subrings (i.e., there is an analogy with groups and semigroups). Moreover, every finite field is both M-periodic and M-primary. Thus, the modification of the concepts discussed, given in [21], is more natural for associative rings.
In [21], properties of M-periodic and M-primary associative rings are studied. In addition, paper [21] characterizes the M-periodic, M-primary, and M-reduced varieties of associative rings. From the results of paper [21] (in particular, Remark 5.17), it follows that the class MC of all M-complete rings of As is closed with respect to homomorphic images, extensions, and direct sums in As. Furthermore, the class MR of all M-reduced rings of As is closed with respect to subrings, direct products, and extensions in As. Besides, the variety As is transverbal (in the sense of [15]) with respect to any variety belonging to M.
If we replace the set At(L(As)) by the set M in paper [18], the main result of [18] will change. From the modified result, we obtain that any ring R belonging to As contains the largest M-complete subring CM(R); CM(R) is a two-sided ideal of the ring R; the factor ring R/CM(R) is an M-reduced ring. All of the above means that the basic properties of the modified concepts of completeness and reducibility for associative rings are saved. Further, let a mapping rM : As ^ As be such that rM(R) = CM(R) for all R € As. From the above, we obtain that rM is a radical in the sense of Kurosh and Amitsur (see, for example, [1, p. 91] or [6, p. 27]). Here, MC is a radical class and MR is a semisimple class.
We say that the radical rM is the M-complete radical and the ideal CM(R) of R is the M-complete radical of the ring R. Note that CM(R) contains any M-complete subring of the ring R. Therefore, rM is a strict radical in the sense of Kurosh [13] (see also [6, p. 148]).
In papers [10, 11, 20, 24, 25], the complete radical of an associative ring was studied. It is easy to verify that analogous main results of these papers also hold for the M-complete radical. In papers [12, 22, 26-28], the structure of complete and reduced associative rings was studied. The main results of these papers are significantly modified if we replace the concepts of completeness and reducibility with the concepts of M-completeness and M-reducibility.
Recall that in the theory of abelian groups, the concept of a complete group coincides with that of a divisible group. Any minimal divisible abelian group is isomorphic to the (additive) quasi-cyclic group , where p is a prime, or to the additive group Q+ of the field Q of rational numbers. Minimal divisible abelian groups have significant importance since any divisible abelian group is a direct sum of minimal divisible abelian groups (see, for example, [4, Theorem 23.1, p. 124]). An associative ring R is called minimal M-complete if it is a nonzero M-complete ring, and all proper subrings of the ring R are M-reduced rings.
This paper aims to characterize M-complete, M-reduced associative Artinian rings, and minimal M-complete finite associative rings. Note that the class of Artinian rings contains all finite rings.
Nevertheless, in this paper, we do not limit ourselves to Artinian rings and provide some results, which are valid for all associative rings. We will use the results from papers [12, 22, 23, 26-29] on the study of complete and reduced associative rings and modify them for the concepts of M-completeness and M-reducibility. If the obtained results have fundamental changes, then proofs are provided. The modified formulations of the statements are given only with references to similar statements if the changes are insignificant. Before formulating and proving the basic results of the paper, we will provide and prove several lemmas. Some lemmas are of independent interest.
First, let us give some definitions, notations, and facts about associative rings.
2. Basic definitions, notations, and preliminary information
Further in the paper, by a ring we mean an associative ring (not necessarily with the unity), by an ideal we mean a two-sided ideal. Denote by |M| the cardinal number of a set M. Positive integers are denoted by k,l,m,n (sometimes with subscripts), and primes are denoted by p and q. Denote by R+ the additive group of the ring R. A ring with zero multiplication will be called an abelian ring. An abelian ring with an additive group R+ is denoted by R0. Denote by O the zero
ideal of the ring R. A simple ring is a nonzero ring having no ideal besides O and itself. The smallest n € N is said to be the characteristic of a ring R if nR = O and is denoted by char R. If there is no such n, then char R = 0 is assumed.
The set of natural numbers is denoted by N, and the set of primes is denoted by P. Denote by Z the ring of integers, and by Q the field of rational numbers. Furthermore, let Zn denote the ring of residue classes modulo n > 1. The finite field (Galois field) of pm elements is denoted by Fpm. A prime field is a field, which has no proper subfields. Any prime field is isomorphic to the field Q of rational numbers or the finite field Fp of p elements.
If M is a nonempty subset of a ring R, then (M) and (M) denote the subring and the ideal of R generated by M, respectively. The subring generated by a € R is called monogenic and denoted by (a). An element e € R with the property e2 = e is called an idempotent of R. The idempotent e € R is called basic if a(e) is the unity of the factor ring R/J(R), where J(R) is the Jacobson radical of the ring R and a is the natural homomorphism of R to R/J(R). A ring R is called idempotent if R2 = R, where R2 = (a ■ b | a, b € R).
Denote by Mn(R) the ring of square n x n matrices over a ring R. For a commutative ring R with unity, R[x] denotes the ring of polynomials in x over R. Denote by Z(X) a free (in As) ring over the infinite countable set X = {x1,x2,... }, i.e., the ring of polynomials with integer coefficients in noncommuting variables of X with zero free terms. An identity is a formal equality of the form f (x1, x2,..., xn) = 0, where f (x1, x2,..., xn) € Z(X).
Suppose that ^ is the natural homomorphism of Zpk [x] onto Zp[x] and f (x) € Zpk [x] is a unitary polynomial of degree m such that <^(f (x)) is an irreducible polynomial over Zp; then the factor ring Zpk [x]/(f (x)) is called a Galois ring of characteristic pk and order pkm. The Galois ring, up to isomorphism, is defined by the numbers p, k, and n and is denoted by GR(pk, m). It is obvious that GR(pk, 1) = Zp k and GR(p, m) = Fpm. Galois rings play a special role in the structural theory of finite associative rings.
Let var£ denote the variety of rings defined by a system £ of ring identities, and let var K be the least variety of rings containing a class K of rings (in other words, var K is the variety generated by K). The free monogenic ring in As will be denoted by Z(x). Below we use the following important notations:
Z0 = var {nx = 0, xy = 0} = var Zn;
Fpm = var {px = 0, xpm = x} = var Fpm.
Let V be a variety of rings. A ring R is called V-complete if R has no homomorphisms onto nonzero rings from V. Equivalently, V(R) = R, where V(R) is the verbal ideal of R (i.e., V(R) is the least ideal in the set of all ideals I of the ring R such that the factor ring R/I belongs to V). A ring is called V-solvable if it has no nonzero V-complete subrings.
Let M be the union of two sets Z and F of varieties of rings, where
Z = {Zp1 | p € P}, F = {Fpm | p € P, m € Z+},
i.e., M = Z U F. Note that M contains the set At(L(As)), where At(L(As)) consists of the varieties Zp and Fp for any prime p (see, for example, [9]). A ring R is called M-complete if R is M-complete for every M € M. We call a ring R M-reduced if R has no nontrivial M-complete subrings. By analogy, the concepts of Z-complete (F-complete) ring and of Z-reduced (F-reduced) ring are defined. We point out the connection between the concepts of completeness and M-completeness, as well as the concepts of reducibility and M-reducibility. Obviously, any M-complete ring is complete. But the converse statement, generally speaking, is incorrect. For example, any nonminimal finite field F is complete, while F is M-reduced. On the other hand, any reduced ring is M-reduced.
Recall that if a variety V is given by an identity system £, then the V-verbal V(R) of a ring R coincides with the ideal of R generated by the values in R of all polynomials that are the left-hand
sides of the identities of £. For varieties Zp and Fpm and a ring R, we indicate formulas to calculate the corresponding verbals:
Z°(R) = pR + R2, Fpm (R) = pR + Rpm, where Rpm is the ideal generated by the set
{rpm - r | r € R}.
It is clear that a ring R is M-complete (M-reduced) if and only if Z°(R) = R (Z°(R) = O) and Fpm (R) = R (Fpm (R) = O) for all p and m, respectively.
It is clear that the M-complete radical CM (R) of a ring R is equal to the sum of all M-complete subrings of R. A ring R is M-complete if and only if CM (R) = R. In particular, the M-complete radical CM(R) of any ring R is an M-complete ideal of R. A ring R is M-reduced if and only if Cm(R) = O. From a well-known fact for arbitrary radicals (see, for example, [1], Proposition 1, p. 91) it follows that the M-complete radical of a ring R is the intersection of all its ideals I such that the factor ring R/I is M-reduced.
Similarly, the Z-complete radical Cz(R) and the F-complete radical Cf(R) of a ring R are defined by the sets Z and F, respectively. Recall that a radical r is called strict if the radical r(R) of a ring R contains every r-radical subring A (i.e., a subring with the property r(A) = A) of R. As above, the M-complete radical is strict. It is clear that Z-complete and F-complete radicals are also strict. A ring is M-complete if and only if it is simultaneously Z-complete and F-complete. Denote by M (3, F) the class of all rings belonging to the varieties of rings from the set M (Z, F), respectively. It is clear that M-complete (Z-complete, F-complete) radicals are upper radicals defined by the class M (3, F), respectively. In addition, for any prime p, we need the notation Fp for the class of rings of characteristic p from the class F.
Recall that the transverbality of the variety As over the subvariety V means that, for any ring R and an ideal I of R, V(I) is an ideal of R. As already noted, the transverbality of the variety As over any variety from the set M is proved in paper [21].
In additive notation, the atoms of the lattice L(Ab) of subvarieties of the variety Ab of all abelian groups Ab are the varieties Ap = var {px = 0} for all primes p. Note that the Ap-completeness of an abelian group A means the validity of the equality Ap(A) = pA = A. Further, the divisibility of an abelian group A is equivalent to its Ap-completeness over all primes p, i.e., the completeness of A. It is well known that in every abelian group A, a divisible subgroup is always a direct summand, in A there is the largest divisible subgroup C(A), and A is the direct sum of its complete and reduced subgroups. Moreover, as noted above, every divisible abelian group is the direct sum of some sets of isomorphic copies of the additive group Q+ of rationales and copies of quasi-cyclic groups Cp~ for some primes p.
Recall that a ring is called a left Artinian ring if any decreasing chain of its left ideals stabilizes. Equivalently, the ring satisfies the minimum condition of left ideals. Further, left Artinian rings will be called Artinian rings. It is well known (see, for example, [8, Theorem 1, p. 63]) that the Jacobson radical of an Artinian ring is nilpotent. In addition, by the Wedderburn-Artin Theorem (see, for example, [8, p. 65]), any Artinian semisimple (in the sense of Jacobson radical) ring is isomorphic to the direct sum of finitely many full matrix rings over skew fields. It is well known that a factor ring of an Artinian ring is Artinian. Also, if the ideal I and the factor ring R/I of a ring R are both Artinian, then R itself is Artinian.
In conclusion of this section, we give some well-known statements that do not relate to the concepts of M-completeness and M-reducibility but are needed for the sequel.
Theorem 1. [5, Theorem 122.7, p. 350] Every Artinian ring R is the ring direct sum R = S ® Tpi ® ■ ■ ■ ® TPk of some torsion-free Artinian ring S and a finite number of Artinian pi-rings Tpi corresponding to various primes pi.
Theorem 2. [32, Proposition 6] Let R be a finite ring with unity of characteristic pk and radical J(R). Then R contains a subring Q isomorphic to a direct sum of matrix rings over Galois rings such that Q/pQ = R/J(R) and a (Q, Q)-submodule M of J(R) such that R = Q + M with Q n M = O.
Theorem 3. [7, Theorem 1.4.3, p. 35] If an additive group R+ of a left Artinian ring R is a torsion-free group, then R possesses a left unity.
Proposition 1. [28, Lemma 12] An Artinian ring R is finite if and only if mR = O for some m € N and the factor ring R/J(R) is finite.
Lemma 1. [10, Lemma 3] If I is an ideal of a ring R and K is afield, then any homomorphism ^ : I ^ K can be extended to a homomorphism : R ^ K.
Lemma 2. [24, Lemma] For any ideal I of a ring R, the relation Mn(R)/Mra(I) = Mn(R/I) holds.
3. Artinian M-complete rings
This section aims to obtain a characterization of M-complete Artinian rings. Let us first give several lemmas. Some of them are valid for arbitrary rings.
Lemma 3. If R is an M-complete ring, then R2 is an M-complete ring.
Proof. Let R be an M-complete ring. Since R is a Zp1 -complete ring for any prime p, the relation ZJ0(R) = pR + R2 = R holds for all primes p; i.e., R = R2 + pR. We have
R2 = (R2 + pR)(R2 + pR) = R4 + pR3 + p2R2 = R4 + pR(R2 + pR) = R4 + pR2;
i.e., Zp(R2) = R2. This equality means that the ring R2 is Z^-complete.
We now show that R2 is Fpm-complete for all p and m. Since R is a Z^-complete ring for any prime p, for each x of R, we find elements a, a», and 6 (i = 1,..., n) of R such that
x = pa + ^ ajbj. i=1
Then,
n pm n / n Pm n
xpm - x = (pa + ^ ajbj) - (pa + ^ a^) = pz + ( ( ^ a^) - ( ^ aA i=1 i=1 i=1 i=1
for some z € R. Therefore, Rpm Ç pR + (R2)pm. It follows that
(Rpm )2 Ç (pR +(R2)pm )(pR + (R2)pm ) Ç pR2 + (R2)pm . Considering this inclusion and the fact that the ring R is Fpm-complete, we get
R2 = (pR + Rpm)(pR + Rpm) Ç pR2 + (Rpm)2 Ç pR2 + (R2)pm = Fpm (R2); i.e., R2 is Fpm-complete for all p and m. □
Corollary 1. If R is a minimally M-complete ring and R2 = O, then R2 = R.
Lemma 4. A simple ring R is either M-complete and does not belong to the set M or M-reduced and isomorphic to a field Fpm for some p and m.
Proof. A simple ring R is a ring with nonzero multiplication, so R € F for some prime p. As is well known, any nonzero ring of any variety from F is a subdirect product of finite fields (see, for example, [30]). □
Lemma 5. Any nil ring R is an F-complete ring.
Proof. The homomorphic image of a nil ring is a nil ring and therefore cannot be a nonzero ring of a variety Fpm for any p and m. □
Lemma 6. [26, Lemma 3] If the ideal I of a ring R is contained in the kernel of any homo-morphism of R onto rings from a variety V of rings, then R is a V-complete ring if and only if R/I is a V-complete ring.
Repeating almost verbatim the proof of Lemma 5 from [11], one can verify the validity of the following statement.
Lemma 7. A nilpotent ring R is M-complete if and only if its additive group R+ is divisible.
We omit the proof of the following statement analogous to Lemma 2 from [26], which corresponds almost verbatim to the proof of that lemma and uses the results mentioned in Section 1 on the Jacobson radical of Artinian rings, semisimple Artinian rings, the structure of divisible abelian groups, and Lemma 7.
Lemma 8. The following conditions are equivalent for a ring R:
n
(1) R+ = 0 Cpoo ;
(2) R is an M-complete abelian Artinian ring;
(3) R is an M-complete Artinian nilpotent ring.
The main result of this section is a modification of two statements of Theorems 1 and 2 of [26].
Theorem 4. An Artinian ring R is M-complete if and only if the following conditions hold for its ideal R2:
(1) R2 is an idempotent Artinian ring and if R2 = O, then
k
r2/j(R2)=0 Mni(Ki),
i= 1
where Ki is a skew field and Mni(Ki) = Fpm for any prime p, m € N, and i = 1,..., k;
(2) if R2 = R, then i
n
R/R Cp w.
j=i j
Proof. Let R be an M-complete Artinian ring. It follows from Theorem 1 that R is the direct sum of its ideals S and T, where S is an M-complete torsion-free Artinian ring and T is an M-complete Artinian periodic ring. Therefore, it is enough to consider both rings separately.
By Theorem 3, the ring S possesses a left unity. Therefore, S2 = S. Since S is an M-complete ring, it follows that the factor ring S/J(S) is an M-complete ring. From the Wedderburn-Artin theorem and Lemma 4, it follows that the factor ring S/J(S) is isomorphic to a direct sum of finitely many full matrix rings over skew fields and does not contain summands isomorphic to a finite field Fpm for any prime p and m € N.
Further, consider a decreasing chain of ideals in T: T 5 T2 5 T3 5 ... Since T is an Artinian ring, we have Tn = Tn+1 for some n; i.e., the ideal Tn is an idempotent ring. Then, T = T/Tn is an M-complete Artinian nilpotent. ring and, by Lemma 8, T is an abelian ring. Hence, xy € Tn for all € T and, therefore, T2 = Tn; i.e., T2 is an idempotent ring. In addition, by Lemma 8,
t/t2 = 0 c j=1
0
poo .
Let us show that the ideal T2 is an Artinian ring. Let I be a left ideal of T2. The group T + is periodic, therfore, for any i € I, there exists m € N such that mi = 0. By Lemma 8,
T/T2 = 0 CpO,
i=1
it follows that there exists ti € T for any t € T such that mti = t in the factor ring T = T/T'2. Since t — mt1 € T2, we have
ti = (t — mt1 + mt1)i = (t — mt1)i + mt1i = (t — mt1)i € I.
This means that I is a left ideal of T.
Thus, T2 is an Artinian and M-complete ring by Lemma 3. Therefore, T2/J(T2) also is M-complete ring. It follows that
k
T 2/J (T2) =0 Mni (Ki),
i=1
where Ki is a skew field and Mni (Ki) = Fpm for any prime p and m € N.
Conversely, let for the ideal R2 of an Artinian ring R, conditions (1) and (2) of the theorem be satisfied. Four cases are possible:
(i) R = R2 = O. Then, R is an M-complete ring by definition.
n
(ii) R = R2 = O and R/R2 =0 . Then, R = R/R2 is an M-complete ring by Lemma 8.
j=1 j
k
(iii) R = R2 = O and R/J(R) = R2/J(R2) = 0 Mni(Ki), where Ki is a skew field and
i=1 i
Mni (Ki) = Fpm for any prime p and m € N. In this case, R is a nonzero idempotent Artinian ring and R/J(R) is an M-complete ring. Then, J(R) is an M-complete ring by Lemma 5. This means that J(R) is contained in the kernel of any homomorphism onto rings from a variety Fpm for any p and m. Then, by Lemma 6, R/J(R) is an F-complete ring if and only if R is an F-complete ring. The ring R = R2 also is Z-complete. Hence, R is an M-complete ring.
n
(iv) R = R2 and R2 = O, where R/R2 = 0 and the ideal R2 is an idempotent Artinian
j=1 pj
k
ring. Then, R2/J(R2) = 0 Mni (Ki), where Ki is a skew field and Mni (Ki) = Fpm for any
i=1
prime p and m € N. The M-completeness of R2 is proved similar to case (iii). The factor ring R/R2 is M-complete by Lemma 8. In this case, it follows from Lemma 2.2 of [21] that the extension of the M-complete ring R2 by the M-complete ring R/R2 is an M-complete ring. Besides, it is known that an extension of an Artinian ring by an Artinian ring is also Artinian.
□
A special case of Theorem 4 is a modification of the result on complete finite rings from [12].
Corollary 2. A finite nonzero ring R is M-complete if and only if the following conditions hold:
(1) R2 = R;
(2) R/J(R) is an M-complete ring and R/J(R) = ®n=1Mni (Fpim;), where m > 1 for all i = 1,2,..., n, mi € N, and pi are primes.
4. Artinian M-reduced rings
This section aims to characterize M-reduced Artinian rings.
The following statements are modifications for M-reduced rings of lemmas for reduced rings from [27]. Their proofs are easy to obtain if we replace the field Fp by the field Fpm for any p and m.
Lemma 9. [27, Lemma 1] For an Artinian nilpotent ring R, the following conditions are equivalent :
(1) mR = O for some m € N;
(2) R is a finite ring;
(3) R is an M-reduced ring.
From Lemma 9, it follows that all nilpotent M-reduced Artinian rings are finite.
Lemma 10. [27, Lemma 1] Any Artinian M-reduced ring has characteristic m > 0.
From Lemma 10 and Theorem 1, it follows that it is sufficient to characterize Artinian M-reduced rings of characteristic pk.
Lemma 11. [27, Lemma 2] Any ring R of characteristic pk, where k € N and p is a prime, is Z0-complete and Fqm-complete for any prime q = p and m € N.
Lemma 12. [27, Lemma 3] A ring R of characteristic pk for a prime p and k € N, is an M-complete ring if and only if the ring R/pR is M-complete.
The following lemma describes the M-complete radical of an Artinian ring of characteristic pk for a prime p and k € N.
Lemma 13. For an Artinian ring R of characteristic pk, where p is a prime and k € N, there exists m,n € N such that the F-complete radical CF(R) = Fpm (R) and the M-complete radical Cm(R) = Fpm (R), where n is the idempotent degree of the verbal Fpm (R); i.e., Fpm (R) — Fpm (R).
Proof. Consider the set of all verbals Fpd (R) of the ring R, where d € N. Note that, for any t € N and x € R,
j=t
xptd - x = (xpd - xj • ^ xptd-pd-t(pd-1), where s = ^ p(t-j)d.
i=0 j=0
This means that if h is divisible by d, then Fph (R) C Fpd (R).
Since R is an Artinian ring, R contains the minimal verbal Fpm (R). At the same time, Fpm (R) C Fpd (R) for all d € N (assuming that this is not the case, we get that the verbal Fpm (R) is not minimal since it contains the verbal Fpi (R) C Fpm (R) n Fpd (R) = Fpm (R), where l is the least common multiple of d and m).
By Lemma 11, the ring R of characteristic pk is Fqt-complete for any prime q = p and t € N. It follows from Lemma 1 that any ideal of an Fqt-complete ring, in particular, the ideal Fpm(R), is also an Fqt-complete ring. From the same lemma and the fact that Fpm (R) C Fpk (R) for all k € N, it follows that Fpm (R) is an Fpk-complete ring for all k € N. Therefore, Fpm (R) is an F-complete ring; i.e., Fpm(R) C CF(R); hence, Fpm(R) = CF(R).
The decreasing chain of ideals Fpm (R) 5 Fpm (R) 5 Fpm (R) 5 ... of the ring R stabilizes at some step n. That is, F^m (R) is an idempotent ring; so it is Z-complete. In addition, F^m (R) is an F-complete ring by Lemma 1. So, F^m(R) C CM(R). Conversely, since CM(R) C Fpm(R), we have CM(R) C Fpm(R). Hence, Cm(R) = CM(R) C Fpm(R) and therefore Cm(R) = Fpm(R). □
Lemma 14. ^ nonnilpotent Artinian ring R of characteristic , where p is a prime and k € N, is an M-reduced ring if and only if R is a finite ring and CF(R) = J(R). /n addition, the factor ring R/J(R) is isomorphic to a finite direct sum of fields F^ for k € N.
Proof. First, we show that CF(R) = J(R). It follows from Lemma 5 that J(R) is an F-complete ring, so J(R) C CF(R). Conversely, by Lemma 13, CF(R) = Fpm(R) for some m € N. Since R is an M-reduced ring, we have CF(R) = Fpn(R) = O for some n € N. Thus, Fpm(R) is a nilpotent ideal; hence, CF(R) = Fpm (R) C J(R).
The factor ring R/J(R) is isomorphic to a direct sum of finitely many full matrix rings over skew fields. Since Fpm (R) = J(R), the factor ring R/J(R) belongs to the variety Fpm. Therefore, each of these summands belongs to the variety Fpm and is isomorphic to F^ for some k € N by Lemma 4. Also, since R/J(R) is a finite ring, R is also a finite ring by Proposition 1.
Conversely, if R is a finite ring, then J(R) is an M-reduced ring by Lemma 9. Hence, the ring R is M-reduced as an extension of the M-reduced ring J(R) by the M-reduced ring R/J(R). □
The following statement, similar to Teorem 2 of [27], describes the structure of Artinian M-reduced rings.
Theorem 5. An Artinian ring R is an M-reduced if and only if R is a finite ring with
n
F-complete radical CF (R) = J(R) and either R = J(R) or R/J(R) = 0 F£i, where Pi is a
i=l "
prime and k € N.
Proof. By Lemma 10, for any Artinian M-reduced ring R, there exists m € N such that mR = O. Let m = p^1 ■ p^2 ■ ... ■ p^ be the canonical representation of the number m. Then, by Theorem 1, the ring R is a finite direct sum of its ideals Ri, where pki R = O for all 1 < i < n.
It follows from the properties of a finite direct sum of rings that the rings Ri for all 1 < i < n are M-reduced Artinian rings. If the ring Ri is nonnilpotent, then it satisfies the conditions of Lemma 14, otherwise Ri satisfies the conditions of Lemma 9. If Ri is nilpotent, then CF(Ri) = Ri by Lemma 5. In each case, Ri is a finite ring and Cf(R) = J(Ri). Thus, R is a finite ring and its F-complete radical
n n n
Cf (R) = 0 Cf(R) = 0 Fpmi (Ri) = 0 J(Ri) = J(R).
i=1 i=1 i=1
Moreover, if R = J(R), then the factor ring R/J(R) is a finite direct sum of ideals isomorphic to finite fields.
Conversely, any finite ring R satisfying the conditions of the theorem is an extension of the M-reduced ring J(R) by the M-reduced ring R/J(R). This means that R is an M-reduced ring. □
5. Minimally M-complete Artinian rings
This section aims to characterize minimally M-complete Artinian rings. Before proving the main result, we formulate analogs of auxiliary statements from [22] and [28] and prove some of them.
The proofs of the following several statements almost verbatim correspond to the proofs of their analogs, so, we omit them.
Proposition 2. [25, Proposition 1] For a basic idempotent e of a nonnilpotent Artinian ring R, CM(eRe) = eCM(R)e.
For an M-complete radical, the requirement that the idempotent e is the basic idempotent of a nonnilpotent Artinian ring A is essential. For example, in the M-complete ring R = M2(Fp), for the idempotent
' 1 0
e '00
the subring eRe = Fp is M-reduced; i.e., CM(eRe) = O. However, eCM(R)e = eRe = O.
Corollary 3. [25, Corollary 2] A nonnilpotent minimally M-complete Artinian ring contains a unit.
Lemma 15. [22, Lemma 11] If any decreasing chain of ideals of a ring R contained in the ideal I of this ring stabilizes at some finite step, then the M-reducibility of the ring R implies the M-reducibility of the ring R/I.
Corollary 4. Any homomorphic image of an Artinian M-reduced ring is an M-reduced ring.
Corollary 5. The homomorphic image of a minimally M-complete finite ring is a minimally M-complete ring.
Lemma 16. [28, Lemma 11] A finite idempotent ring R of characteristic pk is minimally M-complete if and only if R/pR is a minimally M-complete ring.
Lemma 17. [22, Lemma 15] If I is a nilpotent ideal of a ring R of characteristic pk and K is an M-reduced subring of the ring R, then the homomorphic image K in the ring R = R/1 is also an M-reduced ring.
Corollary 6. If R is a minimally M-complete Artinian ring of characteristic pk, then the factor ring R/J(R) is also a minimally M-complete ring.
Lemma 18. [22, Lemma 1] A minimally M-complete nilpotent ring is isomorphic to the ring Q0 or the ring C^ for some prime p.
Lemma 19. [22, Lemma 3] A skew field K of characteristic zero is minimally M-complete if and only if it is isomorphic to the field Q of rational numbers.
Corollary 7. The ring Z of integers and any of its subrings are M-reduced.
Proposition 3. [24, Proposition] For any ring R and n > 1, the ring Mn(R) is M-complete if and only if the ring R is Z-complete.
Corollary 8. For any idempotent ring R and n > 1, the ring Mn(R) is M-complete.
The description of minimally M-complete skew field of prime characteristic p differs significantly from the description of complete skew field of prime characteristic p obtained in Lemma 4 of [22].
Lemma 20. A skew field K of prime characteristic p is minimally M-complete if and only if K is isomorphic to the algebraic closure Fp of the field Fp.
Proof. Let K be the minimally M-complete skew field of prime characteristic p. Then K contains the field Fp that obviously lies in the center of K.
Just as in the proof of Lemma 4 of [22], it can be shown that the existence of an element in K that is transcendent with respect to the field Fp is impossible.
Therefore, all elements of the skew field K are algebraic with respect to the field Fp. It is clear that elements of K are algebraic with respect to a field Fpm for any m > 1. Recall that the field Fpm is M-reduced by Lemma 4. Taking into account the well-known facts that, for any finite field Fq in the ring Fq [x], there exists an irreducible polynomial of any positive degree (see, for example, [14, Corollary 2.11, p. 70]) and an algebraic extension of Fq containing any of its roots is again a finite field, it is easy to understand that K must coincide with the union of a countable infinite strictly increasing sequences of corresponding finite fields, i.e., K is isomorphic to the algebraic closure Fp of the field Fp.
Conversely, if a skew field K of prime characteristic p is the algebraic closure of the field Fp, i.e., A = Fp, then any proper nonzero subring F of K is a finite field Fpm for some m and therefore it is M-reduced. Thus, K is a minimally M-complete ring. □
An analog of Lemma 5 from [22] for the M-completeness also has significant changes.
Lemma 21. The full ring Mn(K) of matrices over a skew field K is minimally M-complete if and only if Mn(K) is isomorphic either to the field Q of rational numbers or the algebraic closure Fp of the field Fp or a ring M2(Fp) for some prime p.
Proof. Let Mn(K) be a minimally M-complete ring of matrices of order n over a skew field K. Being M-complete, the ring Mn(K) does not belong to any variety of M since the rings of the latter are M-reduced.
Let n = 1. In this case, M1 (K) = K. If char K = 0, then K = Q by Lemma 19. If char K = p, then K = Fp by Lemma 20.
Let now n > 1. The ring Mn(K), in this case, contains the skew field K as its proper subring. Due to the minimal M-completeness of the ring Mn(K), the skew field K must be M-reduced. By Lemma 4, being a simple ring, K must be isomorphic to a finite field Fpm for some p and m. It is obvious that the ring Mn(Fpm) for n > 3 contains a proper subring isomorphic to an M-complete ring M2(Fpm), and therefore is not M-minimally complete. Hence, n = 2. If m > 1 then the ring M2(Fpm) contains the M-complete proper subring M2(Fp). Thus, m = 1.
Finally, we show that the ring M2(Fp) is minimally M-complete. Consider any proper nonzero M-complete subring R of a ring M2 (Fp). Being finite, and therefore Artinian, a semisimple factor ring R/J(R), by the Wedderburn-Artin theorem, is a direct sum of a finite number of matrix rings over suitable skew fields. It is clear that, in our case, these skew fields must be finite fields. But then the orders of the matrices included in the decomposition of the ring must be equal to 1. Being a direct sum of M-reduced fields, by Lemma 2.5 of [21], the ring R/J(R) must also be M-reduced. But then it is clear that the subring R is not M-complete. Thus, the ring M2(Fp) has no proper nonzero M-complete subrings; therefore, it is a minimally M-complete ring. □
Note that, in [23], it is indicated that Lemma 5 of [22] describing minimally complete rings Mn(K) of all (n x n)-matrices over a skew field K, in the end, mistakenly states that such is the ring M2(Fp) for any p. That this is not the case follows from the well-known representation of finite fields by matrices (see, e.g., [14], p. 90): elements of a finite field Fpn of order pn can be represented by square matrices of order n over the field Fp. Consequently, the ring M2(Fp) contains a subring isomorphic to the complete field Fp2 and, therefore, is not a minimally complete ring. As a consequence, rings of matrices of the form M2(Fp) for all p should be excluded from the formulations of Lemma 5 and condition (2) of the theorem from [22]. Rings R for which the factor ring R/pR is isomorphic to M2(Fp) also must be excluded from the formulation of condition (3) of the same theorem. The exact formulation of the theorem from [22] is Theorem 4 of paper [29].
Corollary 9. A semisimple Artinian ring is minimally M-complete if and only if it is isomorphic either to the field Q of rational numbers or the algebraic closure Fp of the field Fp or a ring M2(Fp) for some prime p.
Lemma 22. A minimally M-complete finite ring R of a prime characteristic is semisimple by Jacobson.
P r o o f. A ring R satisfying the conditions of Lemma 22 is an algebra of finite dimension over a field Fp. The ring R/J(R) is minimally M-complete by Corollary 5. Therefore, by Corollary 9, R/J(R) is the ring M2(Fp) of square matrices of order 2 over a finite field Fp for some p; i.e., R/J(R) is a central simple algebra. In any case, we get that the algebra is a separable algebra over a field Fp. The field Fp is perfect; therefore, according to the Wedderburn-Maltsev theorem (see, for example, Theorem 13.18 in [3], p. 575), R = J(R) ® S, where S is a subalgebra of R and S is isomorphic to R/J(R). Since the ring R/J(R) is M-complete and R is minimally M-complete, we have R = S; i.e., R is a semisimple ring. □
Corollary 10. In a minimally M-complete finite ring R of characteristic pk, the equality J(R) = pR is valid.
Proof. In the ring R, the ideal pR is nilpotent; therefore, pR C J(R). On the other hand, the ring R/pR is minimally M-complete by Lemma 5 and semisimple by Lemma 22. Hence, J(R) C pR; i.e., J(R) = pR. □
Lemma 23. A minimally M-complete ring R of all matrices of some order over the Galois ring is isomorphic to the ring M2(Zpk) for some prime p and k € N.
Proof. For every Galois ring GR(pk, m), the factor ring
GR(pk, m)/pGR(pm, k) = GR(p, m) = Fpm.
Note, that any Galois ring has a unit and therefore is an idempotent ring. By Corollary 8, the matrix ring Mn(GR(pk, m)) is M-complete for any n > 2. It follows that if R = Mn(GR(pk, m)) is the minimally M-complete ring, then n = 2. By Lemma 2,
R/pR = M2(GR(pk ,m))/M2 (pGR(pk ,m)) = M2(GR(p,m)) = M2(Fpm).
It follows from Lemma 21 that
R/pR = M2(Fp) = M2(GR(p, 1)).
But then
R = M2(GR(pk, 1)) = M2(Zpk).
Conversely, let R = M2(Zpk) for some prime p and k € N. Then, R is a finite ring, for which R2 = R and pkR = O. By Lemma 16, the ring R is minimally M-complete if and only if the ring R/pR is minimally M-complete. Since R/pR = M2(Fp) is minimally M-complete by Lemma 21, we see that the ring R = M2(Zpk) is minimally M-complete. □
The main result of this section is the following modification of Theorem 1 from [22].
Theorem 6. (1) Any minimally M-complete nilpotent ring is isomorphic to the ring Q0 or the ring for some prime p.
(2) A simple ring with unit is minimally M-complete if and only if it is isomorphic to the field Q of rational numbers or the algebraic closure Fp of the field Fp or a ring M2(Fp) for some prime p.
(3) A finite ring is minimally M-complete if and only if it is isomorphic to a matrix ring M2(Zpk) for some prime p and k € N.
Proof. (1) This statement of Theorem 6 is the content of Lemma 18.
(2) Let a simple ring with unit is M-minimally complete. Then, it is an Artinian ring (see, for example, Corollary 4, [2], p. 196). But a simple Artinian ring is isomorphic to the ring of matrices Mn(K) for some skew field K and a natural number n by the Wedderburn-Artin theorem. The rest follows from Lemma 9.
(3) Let R be a finite minimally M-complete ring. It follows that the additive group of the ring R is bounded. Then, by Theorem 1, R is a ring of characteristic pk for some prime p. The ring R is nonnilpotent by Lemma 9. It follows that R is a ring with unity by Corollary 3.
By Corollary 6, the ring R/J(R) is also minimally M-complete. Then, R/J(R) = M2(Fp) by Corollary 9. By Theorem 2, the ring R contains a subring S isomorphic to the direct sum of full matrix rings over Galois rings such that S/J(S) = R/J(R). In the ring S, the equality J(S) = pS
is valid. This means that S/pS = M2(Fp). We obtain that S is a minimally M-complete subring of the ring R by Lemma 16. Therefore, R = S.
Minimally M-complete rings of all matrices of some order over Galois rings are described in Lemma 23. Hence, we get that R = M2(Z„k) for some prime p and k € N. □
6. Conclusion
The paper characterizes associative Artinian M-complete (Theorem 4), M-reduced (Theorem 5), and some classes of minimally M-complete associative Artinian rings (Theorem 6). For an exhaustive description of minimally M-complete Artinian rings, it is necessary to consider the remaining unexplored case of Artinian rings of characteristic pk containing a subring isomorphic to the algebraic closure Fp of the field Fp. As examples show, such rings exist for any prime p and k € N.
Acknowledgements
The preparation of this paper was overshadowed by the death of its first author. The second author dedicates this work to his memory, as a sign of gratitude for many years of fruitful collaboration. Leonid Matveevich Martynov always had an active life position, he was distinguished by high professionalism, decency, and love of life.
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