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\issueinfo{6}{07}{}{2000}
\dateposted{July 19, 2000}
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\PII{S 1079-6762(00)00080-9}
%\def\copyrightyear{2000}
\copyrightinfo{2000}{American Mathematical Society}

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\begin{document}


\title{On non-Spechtianness of the variety of associative rings that
satisfy  the identity $x^{32} = 0$}
\author{A. V. Grishin}
\address{Department of Mathematics, Moscow State Pedagogical University, 
Krasnoprudnaya 14, Moscow, Russia}
\email{markov@mech.math.msu.su}

\date{March 22, 1999}
\commby{Efim Zelmanov}
\subjclass[2000]{Primary 16R10}

\begin{abstract}
In this paper we construct examples of $T$-spaces and $T$-ideals 
over a field of characteristic 2, which do not have the finite basis 
property.
\end{abstract}

\maketitle

Since the end of the sixties and up to date a number of examples of 
algebraic 
objects without the finite basis property have been constructed (for 
groups, 
Lie algebras, and certain other rings close to associative). On the 
other hand,
for the associative rings and several other classes of rings a number 
of positive 
results have been obtained. For  finite rings it is the theorem of Lvov and 
Kruse 
\cite{lvov,kruse}, for algebras over a field of characteristic 0 it is 
Kemer's 
theorem (see \cite{kemer}), and for $T$-spaces over a field of characteristic 
0 
it is the author's result (see \cite{grish1}).

At some stage a majority of mathematicians working in $PI$-rings 
believed that any associative algebra over a sufficiently ``good''
ring (for example, over a field or the integers) has the finite basis 
property.
Nevertheless, in this paper we show that this is not true.

%\noindent{\bf Definition.}
\begin{defnn} Let $k$ be an associative and commutative ring with unity, and
let $F = k \langle x_1, \dots,$ $x_i , \dots\rangle$ 
be a countably generated 
associative free algebra over $k$. Any $k$-submodule $V$ of $F$, closed 
under substitutions that replace $x_i$  by an arbitrary element of 
$F$, is 
said to be a {\em $T$-module of $F$}.
\end{defnn}


In particular, if $k$ is a field, then such a $T$-module is called a {\em 
$T$-space}; if $k = {\mathbb Z}$, then it is called a {\em $T$-group}. 
If a $T$-module 
is also an ideal of $F$, then it is called a {\em $T$-ideal}.

We show that in a free ${\mathbb Z}$-algebra 
${\mathbb Z} \langle x_1 , \dots , x_i , \dots \rangle$  there exist infinite 
ascending 
chains of $T$-ideals. Also it will be shown that in the relatively free 
algebra \- ${\mathbb Z} \langle x_1 , \dots , x_i , \dots \rangle / {( [ x_1 , [ 
x_2 ,  x_3 ] ] )}^T$ 
there exist infinite ascending chains of $T$-groups. We present explicit 
constructions.

\begin{Theorem}\label{T1} Let $k$ be a field of characteristic $2$, and 
let $\Omega$ be the $T$-ideal of $F$ generated by the 
polynomial $[ x_1 , [ x_2 ,  x_3 ] ]$. Then the system of polynomials 
$x_1^2 \cdots x_n^2,$  $n \in \mathbb N$, 
generates a $T$-space of the relatively free algebra $F / \Omega$, which 
does 
not have the finite basis property.
\end{Theorem}

We say that a variety $\mathfrak{M}$ of rings has the Specht property 
if any subvariety of $\mathfrak{M}$ has the finite basis property.

The following statement can be deduced from Theorem 1 (but the proof 
is not trivial).

\begin{Theorem}\label{T2} Let $k$ be a field of characteristic $2$. 
Then the $T$-ideal generated in the free algebra 
$F = k \langle x_1 ,\dots, x_i ,\dots, y_1 , y_2 , z_1 , z_2\rangle$ by the 
polynomials
\begin{equation*}
g_0 = y_1^4 z_1^4 z_2^4 y_2^4 y_1^4 z_1^4 z_2^4 y_2^4   ,\end{equation*}
\begin{equation*}
g_n = y_1^4 z_1^4 x_1^2 \cdots x_n^2 z_2^4 y_2^4 y_1^4
z_1^4 x_{n+1}^2 \cdots x_{2n}^2 z_2^4 y_2^4   ,\end{equation*}
where $n \in {\mathbb N}$, does not have the finite basis property.
\end{Theorem}

\begin{Corollary}\label{C1} Let $A$ be a commutative ring 
which can be mapped homomorphically onto a 
${\mathbb Z}_2$-algebra with unity (for example, $A = {\mathbb Z}$). 
Then the polynomials given in Theorems 1 
and 2 generate a $T$-ideal (\/$T$-module, $T$-group) of $A \langle x_1 , 
\dots,$  $x_i , \dots , y_1 , y_2 , z_1 , z_2 \rangle,$ which is not 
finitely generated.
\end{Corollary}

\begin{Corollary}\label{C2} The variety of associative rings does 
not have the Specht property. Furthermore, the variety of 
associative rings that satisfy the identity $x^{32} = 0$ does 
not have it either.
\end{Corollary}

The details of the proofs of these theorems are given in \cite{grish2} .

\begin{thebibliography}{99}
\bibitem {lvov} I. V. Lvov, {\em Varieties of associative rings. I}, 
Algebra i Logika {\bf 12} (1973), 269--297.
(Russian)
\MR{52:10802}
\bibitem {kruse} B. I. Kruse, {\em Identities satisfied by a finite 
ring}, J. Algebra  {\bf 26} (1973), 298--318.
\MR{48:4025}
\bibitem {kemer} A. R. Kemer, {\em Finite basability of identities of 
associative algebras}, 
Algebra i Logika  {\bf 26} (1987), 597--641; English transl.,
Algebra and Logic  {\bf 26}
 (1987), 362--397. 
\MR{90b:08008}
\bibitem {grish1} A. V.  Grishin, {\em On the finite basis property of 
abstract $T$-spaces}, 
Fundamental'naya i Prikladnaya Matematika   {\bf 1} (1995), 669--700.
(Russian)
\bibitem {grish2} A. V.  Grishin, {\em Examples of $T$-spaces and 
$T$-ideals over a field of characteristic $2$ 
without the finite basis property}, Fundamental'naya i 
Prikladnaya Matematika  {\bf 5} (1999), 101--118. (Russian)
\end{thebibliography}
\end{document}