Equationally noetherian algebras and chain conditions Текст научной статьи по специальности «Математика»

УДК 512.5

Equationally Noetherian Algebras and Chain Conditions

Mohammad Shahryari*

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz,

Iran

Received 26.09.2013, received in revised form 06.10.2013, accepted 06.11.2013 In this article, we describe the relation between the properties of being equational noetherian and ascending chain condition on ideals of an arbitrary algebra. We also give a formulation of Hilbert’s basis theorem for varieties of algebras and obtain a criterion to investigate it for a given variety.

Keywords: algebraic structures, equations, algebraic set, radical ideal, coordinate algebra, Zariski topology, noetherain algebra, equationally noetherian algebra, pre-variety, variety, free product, max-n group, Hilbert’s basis theorem.

Introduction

Universal algebraic geometry is a new area of modern algebra, whose subject is basically the study of equations over an arbitrary algebraic structure A. In the classical algebraic geometry A is a field. Many articles already published about algebraic geometry over groups, see [1-3,11,14], and [16]. In an outstanding series of papers, Z. Sela, developed algebraic geometry over free groups to give affirmative answers for old problems of Alfred Tarski concerning universal theory of free groups (see [18]). Also in [13], two problems of Tarski about elementary theory of free groups are solved. Algebraic geometry over algebraic structures is also developed for algebras other than groups, for example there are results about algebraic geometry over Lie algebras and monoids, see [9,15], and [19]. Systematic study of universal algebraic geometry is done in a series of articles by V. Remeslennikov, A. Myasnikov and E. Daniyarova in [5-8], (during this article, we cite to these papers as DMR-series).

In this article, after a fast review of basic concepts of universal algebraic geometry, we describe the relation between the properties of being equational noetherian and ascending chain condition on ideals of an arbitrary algebra. We also give a formulation of Hilbert’s basis theorem for varieties of algebras and obtain a criterion to investigate it for a given variety. For any algebraic language L and a fixed algebra A of type L, we define the concept of noetherian A-algebra and we prove that the term algebra TL(A)(X) is noetherian if and only if every A-algebra is A-equationally noetherian, provided that A has a trivial subalgebra. Here L(A) denotes the language obtained from L by adding new constant symbols a £ A. We will prove this theorem in a more general setting; we define algebraic geometry over a pre-variety of algebras and prove that the free algebra of a pre-variety of A-algebras is noetherian if and only if every element of that pre-variety is A-equationally noetherian.

The reader who needs some backgrounds of universal algebra, should see books [4,12], or [17]. Our notations here almost the same as in DMR-series. Many results of this work can be stated for structures over any first order language, but for the sake of simplicity, we restrict ourself for the case of algebraic languages.

*[email protected] © Siberian Federal University. All rights reserved

1. Algebraic sets and coordinate algebras

The notations in this article are taken from DMR-series. We need to review some definitions. From now on, L is an arbitrary algebraic language and A is a fixed algebra of type L. The extended language will be denoted by L(A) and it is obtained from L by adding new constant symbols a G A. An algebra B of type L(A) is called A-algebra, if the map a ^ aB is an embedding of A in B. Note that here, aB denotes the interpretation of the constant symbol a in B. We assume that X = (xi,..., xn} is a finite set of variables. We denote the term algebra in the language L(A) and variables from X by TL(A)(X). An equation with coefficients from A is formula of the form p(x1,..., xn) « q(x1,..., xn), where

p(xi, . . . ,xn), q(xi, ...,xn) G TL(a)(X ).

For the sake of convenience, we will denote such an equation by p « q. The set of all such equations will be denoted by AtL(A)(X). Any subset S C AtL(A)(X) is called a system of equations with coefficients from A. A system S is called consistent, if there exists an A-algebra B and an element (b1,..., bn) G Bn such that for all equations (p « q) G S, the equality

pB(bi,...,bn) = qB(bi,... ,bn)

holds. Note that, pB and qB are the corresponding term functions on Bn. A system of equations S is called an ideal, if it is a congruent set on the term algebra, i.e. the set

©s = {(p,q): p ~ q G S}

is a congruence on TL(A)(X). Some times we denote this congruence by the same symbol S. For an arbitrary system of equations S, the ideal generated by S, is the smallest congruent set containing S and it is denoted by [S].

Suppose B is an A-algebra. An element (bi,..., bn) G Bn will be denoted by b, some times. Let S be a system of equations with coefficients from A. Then the set

Vb (S) = (b G Bn : V(p « q) G S, pB (b) = qB (b)}

is called an algebraic set. It is clear that for any non-empty family (Sj}ie/, we have

Vb (U Si) = f| Vb (Si).

iel iel

So, we define a closed set in Bn to be empty set or any finite union of algebraic sets. Therefore, we obtain a topology on Bn, which is called Zariski topology. For a subset Y G Bn, its closure with respect to Zariski topology is denoted by Y. For any set Y, we define

RadB(Y) = (p « q : V b G Y, pB(b) = qB(b)}.

It is easy to see that RadB (Y) is an ideal. Any ideal of this type is called a radical ideal. The coordinate algebra of Y is the quotient algebra

r(Y) = Tl(a)(X )

^ ; RadB (Y )'

An arbitrary element of r(Y) will be denoted by [p]Y.

Suppose Y C Bn and p is a term. Define a function pY : Y ^ B by the rule

pY(b) = pB(bi,... ,bn).

This is called a term function on Y. The set of all such functions will be denoted by T(Y) and it is naturally an A-algebra. It is easy to see that the map [p]Y ^ pY is a well-defined A-isomorphism. So, we have r(Y) = T(Y).

2. Equational Noetherian algebras and chain conditions

An algebra B is said to be equationally noetherian, if for any system of equations S C AtL(X), there exists a finite sub-system So, such that Vb (S) = Vb (So). More generally, an A-algebra B is called A-equationally noetherian, if for any system of equations S C AtL(A)(X), there exists a finite subset S0 C S, such that Vb (S) = Vb (S0). It is easy to show that we can choose S0 from [S], rather than S. It is proved that (see [6]) an A-algebra B is A-equationally noetherian, if and only if every descending chain of closed subsets in Bn has finite length, for any n. Remember that a system of equations said to be an ideal if

©s = {(p, q): p ~ q G S}

is a congruence on TL(A)(X). We say that S is A-ideal if for any distinct ai,a2 G A, we never have ai « a2 G S. Similarly, a congruence R on an A-algebra B, is called A-congruence, if aiRa2, with ai,a2 G A, implies ai = a2. An A-algebra is called noetherian, if it satisfies the ascending chain condition on A-congruences. Note that this implies that every A-congruence is finitely generated and vise versa.

Suppose A contains a trivial subalgebra. Then we claim that the term algebra TL(A)(X) is noetherian, if and only if every A-algebra is A-equationally noetherian. We will prove this assertion in a more general form, for any pre-variety of A-algebras. Hence, we need to define the notions of universal algebraic geometry with respect to a given pre-variety.

In the sequel, we assume that A is an algebra containing a trivial subalgebra. Suppose X is a pre-variety of A-algebras. As before, let X be a finite set of variables. Suppose RX is the smallest A-congruence with the property TL(A)(X)/RX G X. It can be easily shown that

Rx = {(p ~ q) G Aí£(a)(X) : X = Vxi... Vxn(p « q)}.

Let Fx(X ) = M.

RX

It is well-known that FX(X) belongs to X and it is freely generated by the set X. We denote an arbitrary element of FX(X) by p, where p is a term in L(A). Note that if L is the language of groups and X is the variety of all groups, then FX(X) = F(X), the free group with the basis X. If A is a group and X is the class of all A-groups, then FX(X) = A * F(X), the free product of A and the free group F (X).

Suppose now, B G X and b G Bn. We know that there exists a homomorphism : FX(X) ^ B such that

y>(p) = pB(bi,... ,bn).

Therefore, if pi = p2, then pB(bi,...,bn) = pB(bi,...,bn). This shows that the following definition has no ambiguity.

Definition 2.1. An X-equation is an expression of the form p « q, where p and q are terms in the language L(A). If B is an A-algebra and b is an element of Bn, we say that b is a solution of p ~ q, if pB(bi,... ,bn) = qB(bi,..., bn).

Let S be a system of X-equations. The set of all solutions of elements of S, will be denoted by VBí(S). The following observation shows that this is an ordinary algebraic set. Let S' be the set of all equations p « q such that p « q belongs to S. Then it can be easily verified that

Vb (S) = Vb (S').

Therefore, in the sequel we will denote the algebraic set Vb^S) by the same notation Vb (S). The Zariski topology arising from algebraic sets relative to the pre-variety X is the same as the ordinary Zariski topology. If Y C Bn, we define

RadB (Y) = {p « q : Vb Є Y pE(bb ..., bn) = qE (bb ..., bn)}. The quotient algebra

Гх(У )= FX(X >

RadE(Y )

is the X-coordinate algebra of Y. Again, it is easy to see that rX(Y) = r(Y). We are now, ready to prove our main theorem.

Theorem 2.1. Let Y be a variety of algebras of type L and A Є Y containing a trivial subalgebra. Let X = Y A be the class of elements of Y which are A-algebra. Then the free algebra Fx(X) is noetherian if and only if every B Є X is A-equationally noetherian.

Proof. Suppose first that FX(X) is noetherian and B Є X. Let S be a system of X-equations

and ©s = {(p,q): p « q Є S}.

Let R be the congruence, generated by ©s. If R is not an A-congruence, then distinct elements al and a2 in A do exist such that (al, a2) = (al ,a2 ) Є R. Let

S* = {p « q : (p,q) Є R}.

It is easy to see that

Vb (S) = Vb (S*) = 0,

so, we can assume that S0 = {al « a2} Ç S*, and hence Ve (S) = Ve (S0). Therefore, we assume that R is an A-congruence. By assumption, R is finitely generated, so there exists a finite subset R0 Ç R, such that [R0] = R. Let S0 be the set of equations corresponding to R0. Then S0 Ç S* and we have S0* = S*. Hence

Ve (S) = Ve (S*) = Ve (S*) = Ve (S0).

This proves that B is A-equationally noetherian.

Conversely, suppose every B Є X is A-equationally noetherian. Let R be an A-congruence on FX(X ). The algebra FX(X )

Br =“V“

belongs to Y and it is an A-algebra. So, BR Є X. We denote an arbitrary element of BR by t/R. If (p, q) Є R, then

pBR (xi/R, ..., xn/R) = p(xl,..., xn)/R = q(xl,..., xn)/R = qBR (xi/R,..., xn/R).

Hence, if we let SR to be the set of X-equations corresponding to R, then the generic point is a

solution of Sr, i.e. /-/О - (a \

(xl/R, ...,xn/R) Є Увл (Sr).

Suppose p « q is an arbitrary X-equation and

(xi/R, . . . ,xn/R) Є Увд(p « q).

Then, (p, q) Є R. Keeping in mind this observation, now assume that

Rl С R2 С • • •

is a proper chain of A-congruences in Fx(X). For any i, let

Si = {p « q : (p,q) Є Ri}.

Suppose (pi, qi) Є Ri+l \ Ri and Li is the congruence generated by Ri and (pi,qi). Finally, let

Ti be the set of X-equations corresponding to Li. Then we have Si £ Ti Ç Si+l. Let B = Пi Bi,

where Bi = FX(X)/Ri. Then clearly, B Є X and so, it is A-equationally noetherian. Since

A contains a trivial subalgebra, so we can assume that Bj < B, for all i. Now, by the above observation,

(xi/Ri,... ,x„/Rj) e VBi(Si) C Vb (Si),

but it is not an element of VB(Tj) and consequently, it does not belong to VB (Sj+i). This shows that the following chain of algebraic sets is proper

Vb(Si) D Vb(S2) }••• ,

which is a contradiction. □

Now, we are able to give an exact formulation of Hilbert’s basis theorem. Suppose L is an algebraic language and Y is a variety of algebras of type L. Let A e Y and X = YA be the class of all elements of Y which are A-algebra. If A has maximal property on its ideals, is the algebra Fx(X) noetherian?

Example 2.1. Let L = (0,1, +, x) be the language of unital rings and Y be the variety of all commutative rings with unite element. Let A e Y and X = Ya- If X = {x1,...,xn}, then Fx(X) = A[x1,..., xn] and hence Hilbert’s basis theorem is valid in this case.

Example 2.2. Let L = (e,-1, •) be the language of groups. Let Y be the variety of groups. Let A be any group and X = Ya- Then Fx(X) = A * F(X). We show that FX(X) is not noetherian even if A has maximal property on its normal subgroups (max-n). Consider the Baumslag-Solitar group

Bm,„ = (a,t : tamt-1 = a"),

where m,n ^ 1 and m = n. Then, as is proved in [1], this group is not equationally noetherian. Let B = A * Bm,n. Then B is an A-group which is not A-equationally noetherian. So, by the above theorem A * F(X) is not noetherian, Hilbert’s basis theorem fails.

Example 2.3. Let Y be the variety of abelian groups and A e Y be finitely generated. Suppose X = Ya- Then it is easy to see that Fx(X) = A x Fab(X), where Fab(X) is the free abelian group generated by X. So, Fx(X) = A x Z". As a Z-module, clearly A x Z" is noetherian, so Hilbert’s basis theorem is true for any finitely generated abelian group A in the variety of abelian groups. As a result, every abelian group B containing A is A-equationally noetherian.

As we mentioned above, if A < B and B is not equationally noetherian, then it is also not A-equationally noetherian. So, let Y be a variety of algebras and A e Y. Let X = YA. If there exists an element B e Y which is not equationally noetherian, then by our theorem, Fx(X ) is not noetherian, so we never have a version of Hilbert’s basis theorem for the variety Y.

Example 2.4. Let Y be the variety of nilpotent groups of class at most c. If A e Y and X = Ya and B e Y is not finitely generated, then by [16], B is not equationally noetherian and hence Fx(X) is not noetherian.

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Эквационально нетеровы алгебры и цепные условия

Мохамад Шахриари

В этой статье мы описываем соотношение между свойствами эквациональной нетеровости условия восходящих цепей в идеалах произвольной алгебры. Мы также даем формулировку теоремы Гильберта о базисе и получаем критерий в изучении его для данного многообразия.

Ключевые слова: алгебраические структуры, уравнения, алгебраическое множество, радикал, координатная алгебра, топология Зарисского, нетерова алгебра, эквициональность нетеровых алгебр, предмногообразие, многообразие, свободное произведение, максимальная п-группа, теорема Гильберта о базисе.

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