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5
DOI: 10.17516/1997-1397-2020-13-5-583-595 УДК 512.5
On the Equationally Artinian Groups
Mohammad Shahryari*
Department of Mathematics College of Science Sultan Qaboos University Muscat, Oman
Javad Tayyebi^
Department of Pure Mathematics Faculty of Mathematical Sciences University of Tabriz Tabriz, Iran
Received 26.05.2020, received in revised form 02.07.2020, accepted 16.08.2020
Abstract. In this article, we study the property of being equationally Artinian in groups. We define the radical topology corresponding to such groups and investigate the structure of irreducible closed sets of these topologies. We prove that a finite extension of an equationally Artinian group is again equationally Artinian. We also show that a quotient of an equationally Artinian group of the form G[t] by a normal subgroup which is a finite union of radicals, is again equationally Artnian. A necessary and sufficient condition for an Abelian group to be equationally Artinian will be given as the last result. This will provide a large class of examples of equationally Artinian groups.
Keywords: algebraic geometry over groups, systems of group equations, radicals, Zariski topology, radical topology, equationally Noetherian groups, equationally Artinian groups.
Citation: M. Shahryari, J.Tayyebi, On the Equationally Artinian Groups, J. Sib. Fed. Univ. Math. Phys., 2020, 13(5), 583-595. DOI: 10.17516/1997-1397-2020-13-5-583-595.
In the mid-twentieth century, Alfred Tarski asked whether two arbitrary non-abelian free groups are elementary equivalent. To answer this question, it was necessary to investigate systems of equations over groups. Makanin and Razborov proved that the existence of solutions for systems of equations over free groups is a decidable problem and an algorithm to solve such systems of equation is discovered (Makanin-Razborov diagrams, see [10] and [14]). The work of Makanin and Razborov as well as many other mathematicians was the beginning of algebraic geometry over groups. Since then, this new area of algebra was the subject of important studies in group theory. The work of Baumslag, Myasnikov and Remeslennikov provides a complete account of this new subject, [1]. Positive solution to the problem of Tarski is discovered by Kharlampovich, Myasnikov and Sela at the the beginning of the recent century (see [7-9] and [15]). After that, many mathematicians investigated the algebraic geometry over general algebraic systems and this new area of algebra is now known as universal algebraic geometry. The reader can see the works of Daniyarova, Myasnikov, and Remeslennikov as well as the lecture notes of Plotkin as introduction to this branch, [3-6], and [13].
One of the very important notions in the algebraic geometry of groups (as well as other algebraic structures) is the property of being equationally Noetherian. Note that if S is a system of equations over a group A, then we say that the system S implies an equation w k 1, if every
*[email protected] [email protected] © Siberian Federal University. All rights reserved
solution of S in A is also a solution of w « 1. This gives us an equational logic over the group A which is not in general similar to the first order logic. For example, the compactness theorem may fails in this equational logic. There are examples of groups such that the compactness for the systems of equations fails (see [1] and [5] for some examples). In some groups, every system of equations is equivalent to a finite subsystem, such groups are called equationally Noetherian. Free groups, Abelian groups, linear groups over Noetherian rings and torsion-free hyperbolic groups are equationally Noetherian. To see interesting properties of this types of groups, the reader can consult [11] and [16]. This kind of groups have very important roles in algebraic geometry of groups. There are many equivalent conditions for the property of being equationally Noetherian, for example, it is known that a group A has this property, if and only if, for any natural number n, every descending chain of algebraic sets in An is finite. According to this equivalent condition, in [11] and [12], the dual property of being equationally Artinian is defined. A group A is equationally Artinian, if and only if, for any natural number n, every ascending chain of algebraic sets in An is finite. In [12], many equivalent conditions to this property is given.
In 1997, Baumslag, Myasnikov, and Romankov proved two important theorems about equationally Noetherian groups: first, they showed that a virtually equationally Noetherian group is equationally Noetherian. They also showed that quotient of an equationally Noetherian group by a normal subgroup which is a finite union of algebraic sets, is again equationally Noetherian (see [2]). In this Article we prove similar results for the case of equationally Artinian groups. These results will provide a large class of examples for equationally Artinian groups. Also, we study irreducible closed subsets of the radical topology in the case of equationally Artinian groups and we obtain a necessary and sufficient condition for an Abelian group to be equationally Artinian.
1. Preliminaries
Let G be an arbitrary group and suppose that X = {xl,... ,xn} is a finite set of variables. Consider the free product G[X] = G* F[X], where F[X] is the free group over X. Every element w G G[X] corresponds to an equation w « 1, which is called a group equation with coefficients from G. If w = w(xi,..., xn, gl,..., gm) G G[X], then the expression w « 1 is a G-equation with coefficients g\,...,gm G G. Suppose H is a group which contains G as a distinguished subgroup. Then we say that H is a G-group. A tuple h = (hi,..., hn) G Hn is called a root of the equation w « 1, if
w(hi,...,hn,gi,...,gm) = 1.
An arbitrary set of G-equations is called a system of equation with coefficients from G. The set of all common roots of the elements of S in H is called the corresponding algebraic set of S and denoted by VH(S). Clearly, the intersection of a non-empty family of algebraic sets is again an algebraic set but the same is not true for unions of algebraic sets. If we define a closed subset of Hn to be an arbitrary intersection of finite unions of algebraic sets, then we get a topology on Hn, which is known as Zariski topology.
For a subset E C Hn, we define the corresponding radical Rad(E) to be the set of all elements w G G[X] such that every element of E is a solution of w « 1. This is a normal subgroup of G[X] which is called the radical of E and the quotient group r(E) = G[X]/Rad(E) is called the coordinate group of E. Similarly, for a system S, we define its radical to be RadH(S) = Rad(VH(S)). This is the largest system of G-equations equivalent to S over H. The corresponding coordinate group is rH(S) = G[X]/RadH(S). It is proved that the study of coordinate groups is equivalent to the study of Zariski topology, i.e. algebraic geometry of H reduces to the study coordinate groups, [1].
A G-group H is called G-equationally Noetherian, if for every system S, there exists a finite subsystem S0, such that VH(S) = VH(S0). Such G-groups have important role in the study of algebraic geometry over G-groups. There are two extremal cases: if G =1, we say that H is 1-equationally Noetherian or equationally Noetherian without coefficients, and if G = H, then we say that H is equationally Noetherian (or equationally Noetherian in Diophantine sense). It is proved that a 1-equationally Noetherian finitely generated group is equationally Noetherian as well, [1]. The class of equationally Noetherian groups is very large, containing all Free groups, Abelian groups, linear groups over Noetherian rings and torsion-free hyperbolic groups are equationally Noetherian. It is not hard to see that the following statements are equivalent for a G-group H:
i- H is G-equationally Noetherian.
ii- the Zariski topology on Hn is Noetherian for all n.
iii- every chain of coordinate groups and proper epimorphisms
p(Ei) ^ Г(Е2) ^ Г(Ез) ^ •••
is finite.
The authors of [2] proved two important theorems about equationally Noetherian groups. The first theorem shows that a finite extension of an equationally Noetherian group is again equationally Noetherian. The second theorem says that if G is equationally Noetherian and N is a normal subgroup which is a finite union of algebraic sets (in Diophantine case), then G/N is also equationally Noetherian. In this article, we are dealing with the dual notion, the property of being equationally Artinian and we prove the similar statements for this type of groups.
2. Equationally Artinian groups
Equationally Artinian algebras are introduced in [11] and [12]. In this section, we review this notion for the case of G-groups. We say that a G-group H is G-equationally Artinian, if for any n, every ascending chain of algebraic sets in Hn terminates. This is not equivalent to the property of being Artinian for the Zariski topology, instead we define a new topological space which becomes Noetherian if H is equationally Artinian. Suppose
T = {uRad(E) : E С Hn, u € G[X]}.
Note that for arbitrary cosets uRad(E) and vRad(F), if their intersection is non-empty, then for an arbitrary element w € uRad(E) П vRad(F), we have wRad(E) = uRad(E) and wRad(F)= vRad(F). Hence uRad(E) П vRad(F) = w(Rad(E) П Rad(F)) = wRad(EU F). This shows that the intersection of two cosets of radicals, is again a coset of a radical subgroup (or it is empty). The set T is a subbasis of closed sets of a topology on the set G[X] which is called the radical topology on G[X] corresponding to H (this topology is finer than the previous one defined in [12], in fact the subbasis introduced in [12] is a fundamental system of closed sets containing the identity of G[X]). Every closed set in G[X] is an arbitrary intersection of finite unions of cosets of the form uRad(E), with E С Hn and u € G[X]. In [12], it is proved that the following statements are equivalent for a G-group H:
i- H is G-equationally Artinian.
ii- for any n and any subset E С Hn, there exists a finite subset E0 С E, such that Rad(E) = Rad(E0).
iii- the corresponding radical topology over G[X] is Noetherian.
Remark 1. The proof is essentially the same as in [12], but since we used here our enhanced definition of finer radical topology, so we show that why the proof remains unchanged. We only need to show that for a G-group H, being G-equationally Artinian is equivalent to the property of being Noetherian for the corresponding radical topology on G[X]. So, let H be G-equationally Artinian and
T = {uRad(E) : E C Hn, u G G[X]}. We first prove that T satisfies the descending chain condition. Suppose
uiRad(Ei) D u2Rad(E2) 2 u3Rad(E3) D • • •
is a descending chain of elements of T. Then we have also the following chain
Rad(E1) D Rad(E2) D Rad(E3) D • • • .
Therefore,
VH(Rad(E1)) C VH(Rad(E2)) C VH(Rad(E3)) C • • • , and this later chain terminates, as H is G-equationally Artinian. So, for some k, we have
Vh (Rad(Efc)) = Vh (Rad(Efc+i)) = Vh (Rad(Efc+2)) = • • • .
Taking one more radical, we get
Rad(Efc) = Rad(Efc+i) = Rad(E^) = • • • .
This shows that
uk Rad(Efc) = ufc+iRad(Efc+i) = ufc+2Rad(Efc+2) = • • • ,
and hence T satisfies the descending chain condition. Now, let Ti be the set of all finite unions of elements of T and T2 be the set of all arbitrary intersection of elements of Ti . Note that T2 is the set of all closed subsets of G[X] with respect to the radical topology. We show that Ti also satisfies the descending chain condition. Suppose that
Mi = uiRad(Ei) U ... U umRad(Em), M2 = wiRad(Fi) U ... U vkRad(Fk)
are sets in Ti and M2 C Mi. For every i ^ m and j ^ k, we have ujRad(Ej) n vjRad(Fj) C ujRad(Ej). Hence we can gain a tree with root vertex ujRad(Ej) and with a unique edge from the root to every proper subset ujRad(Ej) n vjRad(Fj) C ujRad(Ej). Suppose there exists a strictly descending chain of subsets in Ti:
Mi D M2 D M3 D ••• .
As we mentioned, we obtain a tree for any inclusion Mi D Mi+i, such that each vertex is a finite intersection of sets in T, hence each vertex is in T itself, since as we saw above, the non-empty intersections of a finite number of elements from T are again belong to T. Since each vertex is connected to only finite number of other vertices, so each vertex has finite degree. So, every path corresponds to a strictly descending chain of radicals and since H is G-equationally Artinian, so the path is finite. By the well-known Konig's lemma of graph theory, this implies that the graph is finite. Therefore the above chain is also finite. So Ti satisfies the descending chain condition and is closed under finite intersection.
Now, we prove that T2 satisfies the descending chain condition too. Suppose p|°=i Ri is an infinite intersection of elements of Ti . Then we have the following chain:
Ri D Ri n R2 D Ri n R2 n R3 D • • • .
Since T1 satisfies descending chain condition and is closed under finite intersection, so the chain terminates. Therefore
3k : R1 n R2 n ... n Rk = P| Ri.
i=i
Hence, every infinite intersection of subsets of T1 is in fact a finite intersection in T1 and so it belongs to T1. Consequently, we have T2 = T1 and hence it satisfies the descending chain condition. This shows that the radical topology on G[X] is Noetherian. The proof of the converse statement is trivial.
By (EA)o, we denote the class of all G-equationally Artinian G-groups, by (EA)1, the class of 1-equationally Artinian groups and EA will be used for the class of Equationally Artinian groups (Diophantine case where G = H). In this article, we first prove the following theorem.
Theorem 1. Let G € EA be torsion-free and E C Gn be an algebraic set. Then the set Rad(E) is irreducible and all irreducible closed subset of G[X] is a coset of some radical.
Our main tool to prove this result is a well-known theorem of B. Neumann which says that if a group covered by a finite set of cosets of subgroups, then at least one of those subgroups has finite index. This result of Neumann also will be used to prove the following result.
Theorem 2. Let G € EA be torsion-free and E C Gn be a non-empty algebraic set with Rad(E) = G[X]. Then the interior of Rad(E) is empty.
Note that every Noetherian topological space has finite number of irreducible components. In the case of a torsion-free equationally Artinian group G, the space G[X] has a unique irreducible component, say G[X] itself. Theorem 2, also shows that if G € EA is torsion-free, then G[X] is connected. We will prove the converse for coefficient-free case.
Theorem 3. Let G € (EA)1. Then G is torsion-free if and only if, F[X] is connected.
Note that there are many equationally Noetherian groups which are not equationally Artinian, for example, the additive group Q/Z, the multiplicative group of complex numbers, the quasi-cyclic groups (see also Theorem 8). Many other groups like non-Abelian free groups and torsion-free hyperbolic groups are failed to be equationally Artinian (as they are domains and every equationally Artinian domain is finite). It must be said that, at the time of writing this paper, we don't know if there is equationally Artinian group which is not equationally Noetherian. But, both classes are included in a larger class of groups which we call equationally semi-Noetherian. A group G has this property, if for every system of equations S C G[X], almost every finite subset T C S can be omitted solving the system over G, i.e. there exists a finite subset So C S such that for all other finite subset T C S \ S0, we have VG(S) = VG(S \ T). Clearly, every equationally Noetherian group has this property. We will prove,
Theorem 4. If G € EA, then G is equationally semi-NoeAherian.
Our next theorem concerns about an important relation between the classes (EA)1 and (EA)g. We prove,
Theorem 5. Let G be a finitely generated group and let H be a G-group. If H € (EA)1, then H
€ (EA)g, and as a result, any finitely generated element of (EA)1 is equationally Artinian.
In our sixth theorem, we deal with finite extensions of equationally Artinian groups. We prove,
Theorem 6. Let a group A contains a finite index subgroup H which is equationally Artinian. Then A is also equationally Artinian.
This theorem enables us to conclude that any virtually finitely generated Abelian group is equationally Artinian as well as any finite extension of the additive group of any field. This gives us a large class of examples of such groups. This theorem is EA-version of the similar theorem in [2].
Note that the quotient of an equationally Artinian group is not necessarily equationally Artinian (for example the group Q/Z), but, there exists an important situation, the quotient in which, has this property. Our nest result concerns with these situations. Note that in this theorem, we use the group G[t] = G * (t).
Theorem 7. Let G be an arbitrary group such that G[t] is equationally Artinian. Let R be a normal subgroup of G[t] which is closed in the radical topology of G[t]. Then G[t]/R is also equationally Artinian.
Finally, we will show that an Abelian group G is equationally Artinian, if and only if, it has finite number of periods: let p(G) be the set of orders of torsion elements of G. We will prove,
Theorem 8. An Abelian group G is equationally Artinian, if and only if, p(G) is finite.
3. The proofs
Proof. (Theorem 1 and 2) Suppose G is equationally Artinian. Let Y be an irreducible closed subset of G[X]. Since Y is a finite union of cosets of the form uRad(E), so W = uRad(E), for some algebraic set E C Gn and u e G[X]. Now, for an algebraic set E, we show that Rad(E) is irreducible. Note that every closed subset of Rad(E) has the form ^iRad(Li) U ... U vpRad(Lp), where vi e G[X] and E C Li. Now, if Rad(E) can be written as a union of two closed subsets, then we have
m
Rad(E) = y uiRad(^i),
i=i
for some elements ui e G[X] and algebraic sets Ki with E C Ki. It is a well-known theorem of B. Neumann which says that if a group is covered by a finite number of cosets of subgroups, then at least one of those subgroups has finite index. So, we have for example [Rad(E) : Rad(K1)] < to. Suppose now that G is torsion-free and Rad(K1) = Rad(E). Choose an element w e Rad(E), such that for some a e K1, we have w(a) = 1. Then, for all non-zero integers k we have also wk(a) = 1 and hence all cosets wjRad(K1), (1 < j), are distinct. This shows that Rad(E) = Rad(K1) and so Rad(E) is irreducible. Note that in any Noetherian space, there is a finite number of maximal irreducible sets (irreducible components) and in the case of G[X], Rad(0) = G[X] is the only irreducible component.
Now, we show that the interior of Rad(E) is empty for any E = 0. Let an open set G[X] \ ym=1 wjRad(Ej) be contained in Rad(E). Then we have
m
G[X] = Rad(E) U |J wjRad(Ej),
j=1
and again using the theorem of Neumann, some of these subgroups has finite index, which is shows that G[X] \ Uj=1 wjRad(Ej) = 0. Hence Rad(E) has empty interior. □
Proof. (Theorem 3) In this proof, we denote the coefficient-free radical of a subset E e Gn by Rad0(E), i.e.
Rad0(E) = {w e F[X] : Va e E w(a) = 1}. - 588 -
Suppose first that F[X] is not connected. Then we have
m
F[X] = y uiRad°(Ei),
i=i
for some elements ui G F[X] and subsets Ei C Gn, where m ^ 2 is minimal. Again by the theorem of Neumann, there is an index i such that [F[X] : Rad0(Ei)] is finite and is not equal to 1 by the minimality of m. This shows that the coefficient-free coordinate group r(Ei) is finite and non-trivial. But, we know that this coordinate group embeds inside a direct power of G. So, G is not torsion-free, a contradiction.
Conversely, assume that G is not torsion-free. Let a G G be a non-trivial element of finite order m and put a = (a, 1, . . . , 1) G Gn. Let w = x\. Then clearly, wm G Rad0 ( a). Consider the subgroup (w }Rad° ( a). This subgroup contains all elements x1,... ,xn, and so we have F[X] = (w} Rad° (a). Now, we have
m— 1
F[X] = U wiRad°(a),
and hence, F[X] is not connected.
□
We now prove Theorem 4. Note that the proof can be applied for arbitrary algebraic structures as well.
Proof. (Theorem 4) Suppose G is equationally Artinian and S С G[X] is an infinite system. For simplicity, assume S = {v1,v2, v3,...}. We have the ascending chain
VG(S) = VG(vi, v2 ,v3,...) С VG(v2, v3, vA,...) С VG(v3,v4, v5,...) С ••• .
This chain terminates as G is equationally Artinian, so there exists k such that
VG(vk,vk+1,vk+2, ...) = VG(vfc + i,vfc+2,vfc+3, .. .) = VG(vk+2,vk+3,vk+4, ...) •• • .
This shows that
П VG(vj )= П VG(vj )= П VG(vj ) = ••• , j>k j>k+1 j>k+2
and hence
VG(S) = VG(«1 ,...,vk-i) ^ VG(vj)
= VG(v1 ,...,vk-1) n P| VG(vj) j^k + 1
= Vg(v1 , ... , vk-1) n p VG(vj).
j^k+2
In other words, this argument shows that the algebraic sets VG (vj) can be drop in the intersection for j > k. Let S0 = [v17..., vk-1}. Then by this argument, for any finite subset T C S \ S0, we have Vg(S) = Vg(S \ T). □
Proof. (Theorem 5) Let a1,...,ak be a finite set of generators for the group G. Suppose E C Hn. We prove that there exists a finite subset E0 C E, such that RadG(E) = RadG(E0) (note that, here RadG denotes the radical with coefficient in G). Let S = RadG(E) C G[X]. Every element of S has the form
w = w(x1, . . .
, Xni a1 )...■) ak).
We replace every coefficient ai by a new variable yi, and then a coefficient-free system of equations S(x,y) appears. Let T = E x {(ai,..., ak)} C Hn+k. Now, since H £ (EA)i, so there is a finite subset T0 C T, such that Radi(T) = Radi(To). Clearly, we have T0 = E0 x {(ai,..., ak)}, for some finite subset E0 C E. Obviously, S(x,y) C Radi(T). Let u(x,y) £ Radi(T). Then for all e £ E, we have u(e,a) = 1, so u(x,a) £ RadG(E), and therefore u £ S(x,y). This proves that S(x, y) = Radi(T), and hence S(x,y) = Radi(T0).
Now, we show that S(x,a) = RadG(E0). Suppose w(x,a) £ S(x,a). For any e £ E0, we have w(e,a) = 1, so w(x,a) £ RadG(E0). Conversely, if w(x,a) £ RadG(E0), then for w(x,y) £ Radi(T0) = Radi(T), and this shows that w(x,a) £ RadG(E) = S(x,a). This proves that RadG (E) = RadG (E0) and hence H £ (EA)G. □
Theorem 5, enables us to prove that every finitely generated Abelian group belongs to the class EA (we also can deduce this from Theorem 8). Here we give an elementary proof which shows the infinite cyclic group is equationally Artinian.
Lemma 1. Let H = (a) be infinite cyclic group. Then H is equationally Artinian.
Proof. We first show that H £ (EA)i. Let E C Hn. Every element of E has the form e = = (aj1,..., ajn) for some integers ji,... ,jn. Let w = x^1 x^2 ... xan £ Radi(E). Then w(e) = 1 and hence ajiai+ = 1. This shows that
Radi(E) = P| {xl1 xi?2 ...xan : (aj1 ,...,ajn) £ E,jiai + ••• + jnan = 0}.
Suppose
„ ,, „i1) ,-(1K , ,(2) ,-(2K ..(3) .(3)
E = {(aj1 ,...,ajn ), (aj1 ,...,ajn ), (aj1 ,...,ajn ),...}. Suppose S is the following set of equations
j(i)ai + ••• + j(t)an = 0, (t =1, 2, 3,...).
Since the additive group Z is equationally Noetherian, so there exists a finite subset S0 C S, such that VZ(S) = VZ(S0). Suppose S0 consists of the equations
j (i)a i + • • • + j^an = 0 (t = 1, 2,...,m).
Let E0 = {(a. ),..., oP™1), (aj1 },..., aj<(^)),..., (aj( },..., ajn ))}. Then we have obviously, Radi(E) = Radi(E0). This shows that H is 1-equationally Artinian and hence by Theorem 2, it belongs to EA. □
Now, we show that any direct product of finitely many element of (EA)1 is again in (EA)1. This will prove that every finitely generated Abelian group belongs to (EA)i and hence to EA.
Lemma 2. Suppose A and B are equationally Artinian (1-equationally Artinian). Then so is A x B.
Proof. For a number n and a subset E C (A x B)n, suppose that
E = {ci = (ui, u2,..., un) : i £ I}, where I is an index set. We have up = (aj, bj), for some aj £ A and bp £ B. Now, let
T = {ti = (ai, al2, .. ., an) : i £ I},
and
S = {Si = (bi,b2,...,bin) : i £ I}.
Since A and B are equationally Artinian, so there are two finite subsets T0 C T and S0 C S, such that
RadA(T) = RadA(T0), RadB(S) = RadB(S0).
Suppose for example T0 = [ti,... ,ti} and S0 = {si,..., sk} and k > l. Suppose ti = (a\,..., aln) and si = (bii,...,bin). Using these elements, we can define a finite subset
E0 = [ci = ((ai,bi),..., (an,bn):1 < i < l}, such that Rad(E) = Rad(E0). This shows that A x B is equationally Artinian. □
Summarizing, we have Corollary 1. Every finitely generated Abelian group is equationally Artinian.
There are also infinitely generated Abelian groups which are equationally Artinian: let K be a field and consider its additive group H = (K, +). Every equation with coefficient in K has the form ai xi + ■ ■ ■ + anxn = b for some elements ai,... ,an G Z,b G K, so the corresponding algebraic set is an affine subspace of Kn. This shows that every ascending chain of algebraic sets terminates and hence H is equationally Artinian. However, some Abelian groups are not equationally Artinian. For example, consider the additive group H = Q/Z. Let
E = ^ —+ Z : p = prime\ C H1. >P )
{;
If w(x) = mx + b + Z G Rad(E), then for any prime p, we have —+ Z^ = Z, and this
ma
means that for any prime p,--+ — G Z, which is not true. Another example is the quasi-cyclic
pb
groups G = Zp^, for prime numbers p. This is because, the ascending chain of algebraic sets Va(xp « 1), (n > 1) does not terminate (this fact will be used in the proof of Theorem 8).
Before proving Theorem 6, we introduce some notations from [2]. Let a group A be the semidirect product of a finite subgroup T and a normal subgroup H. Assume that T = [ti = 1, t2,... ,tk}. Let w(xi,..., xn, gi,..., gm) be a group word with coefficients in A and v G An. We can express v uniquely in the form v = (sihi,..., snhn) with si G T and hi G H. We also have gi = ribi for unique elements ri G T and bi G H. Define the map A : An ^ Tn by A(v) = (si, ...,sri) and
w(xi, ...,xn)= w(xi, ...,xn,ri,..., rm).
Note that w is an element of T[X] which depends only on w. For any 1 ^ i ^ n and 1 ^ j ^ k, define hij = t-ihitj G H. Denote the tuple
(hii,. .., hik,. .., hni, .. ., hnk)
by v'. Consider the new variables yij for 1 < i < n and 1 < j < k. In [2], it is proved that there exists a unique element
w'v G H [yii ...,yik, . . . ,yni, .. . ,ynk ],
such that w(v) = w(A(v))wV(v'), and wV depends only on the value of A(v). As a result, it is shown that v G An is a root of w « 1, if and only if, A(v) is a root of w « 1 and v' is a root of w'v « 1. We are now ready to prove Theorem 6.
Proof. (Theorem 6) Replacing H by its core, we can suppose that H is a normal subgroup of A with finite index. Let T = A/H. Then A embeds into the wreath product HI T. Recall that this wreath product is the semidirect product of T and HTL We know that (Lemma 2), HT1 is equationally Artinian and any subgroup of an equationally Artinian group is again equationally Artinian. So, it is enough to prove our theorem using the further assumption A = TH, with T finite, H normal and T n H = 1. We will use all the above notations.
Suppose E C An is an algebraic set and S = RadA(E). We must show that there exists a finite subset E0 C E, such that RadA(E0) = S. Let S = {w : w G S} (see the above discussion). Suppose
Vt(S) = [vi,...,vd}C Tn.
For any 1 < i < d, put Li = VH(S'v.) C Hnk. Here S'v. denotes the set of all w'v., such that w G S. Define also
Ki = [h G Hn : (h)' G Li} C Hn.
We have (Ki)' C Hnk and since H is equationally Artinian, there exists a finite subset K0 C Ki, such that
RadH ((K?)') = RadH ((Ki)').
Assume that E0 = ud=iviK° C An. We show that E0 C E. Let vh G E0. Then h G Ki and hence
S (A(vih)) = S(vi) = 1,
and
Svi((vih)') G S'v^(Li) = 1.
This means that vih G Va(S) = E. Therefore E0 C E.
Now, we claim that RadA(viKi0) = RadA(viKi). To prove this claim, assume that w belongs to the left hand side. Then w(viK°) = 1 and hence w'((viK0)') = 1. This shows that w'v. G RadH((viK0)'). Recall that, by the definition of the map v ^ v', we have (viK0)') = (K0)' and hence w'v. G RadH((K0)') = RadH((Ki)') = RadH((viKi)'). Therefore, for any h G Ki, we have w'vi ((vih)') = 1, and since in the same time w(A(vih)) = 1, we have w(viKi) = 1. This proves the claim.
We now, prove that RadA(E0) = RadA(E). Let w be an element of the left hand side and v G E. We have S(v) = 1 and
d
w G p| RadA(viK°).
i=i
Note that v = A(v)h, for some h G Hn. We have S(A(v)) = 1, so there is an index i such that A(v) = vi. Therefore, v = vih. On the other side, since Sv(V') = 1, so
1 = sv (v') = svi ((vih)').
Hence, (vih)' G Li, and therefore h G Ki. Now, by the above claim, we have
w G RadA(viKi0) = RadA(viKi),
and hence w(v) = 1. This shows that w G RadA(E). □
Theorem 6 shows that any virtually finitely generated Abelian group is equationally Artinian as well as any finite extension of the additive group of any field. This gives us a large class of examples of such groups. We now come to Theorem 7. Note that the similar theorem ([2]) for the equationally Noetherian case deals with the Zarizki topology of Gi and its closed normal subgroups. The dual case here deals with the radical topology of G[t] and its closed normal subgroups.
Proof. Assume that
m
R = URadG(Ki),
i=i
where Ki C G. Note that G is equationally Artinian as G[t] is so. Hence every Ki can be chosen finite. Let H = G[t]/R be not equationally Artinian. Hence there exists a number n and a subset E £ Hn such that Rad# (E) = Rad#(E0), for any finite subset E0 C E. Assume that e0 £ E is an arbitrary element. As Rad#(E) = Rad#({e0}), there exist elements fi £ Rad#({e0}) and ei £ E, such that fi(ei) = 1. Similarly, we have Rad#(E) = Rad#({e0,ei}), so there exist elements f2 £ Rad#({e0,ei}) and e2 £ E, such that f2(e2) = 1. Repeating this argument, we obtain two infinite sequences
fi,f2,fs,... £ H[X],
e0, ei, e2, ... £ E,
such that for any i, fi(e0) = fi(ei) = ••• = fi(ei-i) = 1, but fi(ei) = 1. Note that, here X = {xi,..., xn} and so every element of H[X] is a word in t and elements of X with coefficients in G. Suppose q : G[t, X] ^ H[X] is the canonical map sending elements of G to their cosets, and fixing elements of X and the element t. Suppose also that ^ : (G[t])n ^ Hn is the map
^(ui,..., un) = (uiR,..., unR).
Choose a pre-image f i for fi under q and a pre-image ei for ei under Hence, we have fi £ G[t,X] and ei £ (G[t])n. For any i, we have fi(e0) = 1, so fi(e0) £ R. This shows that, there exists an infinite sequence of numbers
ii(0) < i2(0) < is(0) < ••• ,
and a number 1 < p0 < m, such that
7 i1(0)(e0 ),f i2(0)(e0 ),7i3(0)(e0 ), ... £ RadG (Kpo ).
Equivalently, this shows that for all s, we have
fis (0) £ RadG[t] (e0 (Kp0 )).
By a similar argument, we obtain an infinite subsequence of {is(0)} of the form
ii(1) < ¿2(1) <is(1) < ••• , and a number 1 < pi < m, such that for all s, we have
fis (i) £ RadG[t] (ei (KP1 )).
We continue this process to find an infinite subsequence
ii(k) < i2(k) < is(k) < ••• , of the previous sequence, and a number 1 < pk < m, such that
fis(k) £ RadG[t] (ek(Kpk )),
for all s. Note that all sets ei(KPi) are finite as Ki's are finite. Let
K = U ei(Kpi) C (G[t])n.
i=0
By assumption, G[t] is equationally Artinian, so there exists an index l, such that
i
RadG[i](K) = RadG[t](U ei{KPi)).
¿=0
Assume that j > l. Then for any s, we have
i
fis{j) G P| RadG[i]€i(KPi) = RadG[i](K). i=i
Suppose k = i1(j). Then fk G RadG[t](ek(Kpk)), and hence fk(ek) G RadG(Kpk) C R. This shows that fk (ek) = 1, a contradiction. Hence H is equationally Artinian. □
Finally, we give a proof for Theorem 8.
Proof. (Theorem 8) We first, show that a divisible Abelian group G is equationally Artinian, if and only if, it is torsion-free. Recall that a divisible Abelian group has the form G = Q1 © Y/peJ Zp~, for an index set I and a set J of prime numbers. If G is torsion-free then G = Q1, and since the additive group of rationales is equationally Artinian, so is G. Now, suppose that G is equationally Artinian but is not torsion-free. Then for some prime p, we have Zp~ ^ G, and this implies that Zp~ is equationally Artinian, a contradiction.
Now, suppose that G is an arbitrary Abelian group. Assume that p(G) is finite. We know that G = Tor(G) © G1, where Tor(G) is the torsion part of G and G1 is a torsion-free subgroup. We know that G1 can be embedded in some divisible Abelian group and hence it is equationally Artinian. The torsion part has finite exponent and hence can be written in the form Tor(G) = 0m£p(G) Zm™, where for all m G p(G), an index set Im is associated. Clearly, every component Z^1 is equationally Artinian and since p(G) is finite, so the direct sum is also so. This shows that G G EA.
Finally, suppose that in a group G, the set p(G) is infinite. Let m1 < m2 < m3 < ■ ■ ■ be elements of p(G) such that for all i the integer m1m2 ... mi_1 is not divisible by mi. For any i, assume that ai is an element of order mi. Consider the ascending chain
VG(xmi « 1) C Vg(xmim2 « 1) C Vg(xmim2m3 « 1) C ■ ■ ■ .
This chain does not terminate, because for any i, we have
a1 ,...,ai G VG(xmi-mH « 1),
but ai+1 does not belong to it. Therefore G is not equationally Artinian. □
References
[1] G.Baumslag, A.Myasnikov, V.Remeslennikov, Algebraic geometry over groups, I. Algebraic sets and ideal theory, J. Algebra, 2191999, 16-79.
[2] G.Baumslag, A.Myasnikov, V.Romankov, Two theorems about equationally Noetherian groups, J. Algebra, 1997, 194, 654-64.
[3] E.Daniyarova, A.Myasnikov, V.Remeslennikov, Algebra and Discrete Mathamatics, 1(2008), 80-112. DOI: 10.1142/9789812793416_0007
[4] E.Daniyarova, A.Myasnikov, V.Remeslennikov, Algebraic geometry over algebraic structures, II: Fundations, J. Math. Sc. 185(2012), no. 3, 389-416.
[5] E.Daniyarova, A.Myasnikov, V.Remeslennikov, Algebraic geometry over algebraic structures, III: Equationally noetherian property and compactness, South. Asian Bull. Math., 35(2011), no. 1, 35-68.
[6] E.Daniyarova, A.Myasnikov, V.Remeslennikov, Algebraic geometry over algebraic structures, IV: Equatinal domains and co-domains, Algebra and Logic, 49(2011), no. 6, 483-508.
[7] O.Kharlampovich, A.Myasnikov, Tarski's problem about the elementary theory of free groups has a psitive solution, E.R.A. of AMS, 4(1998), 101-108.
[8] O.Kharlampovich, A.Myasnikov, Irreducible affine varieties over a free group. I: irreducibility of quadratic equations and Nullstellensatz, J. Algebra, 200(1998), no. 2, 472-516.
[9] O.Kharlampovich, A.Myasnikov, The elemntary theory of free non-abelian groups, J. Algebra, 302(2006), 451-552.
10] G.Makanin, Equations in free groups, Math. USSR-Izv., 21(1982), no. 3, 483-546.
11] P.Modabberi, M.Shahryari, Compactness conditions in universal algebraic geometry, Algebra and Logic, 55(2016), no. 2, 146-172.
12] P.Modabberi, M.Shahryari, On the equational Artinian algebras, Siberian Electronic Mathematical Reports, 13(2016), 875-881.
13] B.Plotkin, Seven lectures in universal algebraic geometry, preprints, Arxiv.
14] A.Razborov, On systems of equations in free groups, Math. USSR-Izv., 1982, 25, no. 1, 115-162.
15] Z.Sela, Diophantine geometry over groups: I-X, preprints, Arxiv.
16] M.Shahryari, A.Shevliyakov, Direct products, varieties, and compacness conditions, Groups Complex, Cryptol, 9(2017), no. 2, 159-166.
Об эквивалентно артиновых группах
Мохаммед Шахриари
Математический факультет, Колледж Науки Университет Султана Кабуса Мускат, Оман
Джавад Тайеби
Кафедра чистой математики, Факультет математических наук
Университет Тебриза Тебриз, Иран
Аннотация. В этой статье мы изучаем свойство быть артиновым в группах. Определяем радикальную топологию, соответствующую таким группам, и исследуем структуру неприводимых замкнутых множеств этих топологий. Докажем, что конечное расширение уравновешенно арти-новой группы снова уравновешенно артиново. Мы также показываем, что частное от артиново-уравновешенной группы вида G[t] по нормальной подгруппе, являющейся конечным объединением радикалов, опять-таки уравновешенно артново. В качестве последнего результата будет дано необходимое и достаточное условие, чтобы абелева группа была эквивалентно артиновой. Это обеспечит большой класс примеров уравновешенно артиновых групп.
Ключевые слова: алгебраическая геометрия над группами, системы групповых уравнений, радикалы, топология Зариского, радикальная топология, нетеровы группы, эквационально артиновы группы.
