View source for Variety of rings

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A class of rings $\mathfrak M$ satisfying a given system of polynomial identities. A variety of rings can be defined axiomatically as a hereditary class of algebras that is closed with respect to taking homomorphic images and complete direct sums (see [[Algebraic systems, variety of|Algebraic systems, variety of]]). Since the totality of polynomial identities that are satisfied in a given ring forms a fully-characteristic ideal (a [[T-ideal|$T$-ideal]]) of a free ring, there exists a one-to-one correspondence between varieties of rings and the $T$-ideals of a countably-generated free ring. If for two varieties of rings there is an inclusion $\mathfrak N\subseteq\mathfrak M$, one says that $\mathfrak N$ is a subvariety of $\mathfrak M$. The variety corresponding to the $T$-ideal of identities of the ring $A$ is called the variety generated by the ring $A$. Every variety of rings is generated by a  "universal object"  of it; this is a free ring of the given variety containing an infinite free system of generators: Every mapping from the set of free generators into an arbitrary ring of the variety can be extended to a homomorphism.

Let $M_n$ be the variety generated by the algebra of square matrices of order $n$. For each variety of associative rings of characteristic zero (that is, rings whose additive group is torsion-free) there exists a positive integer $n=n(\mathfrak M)$ such that $M_n\subseteq\mathfrak M$, but $M_{n+1}\not\subseteq\mathfrak M$. A variety of rings is called a Specht variety if every ring in it has a finite basis of identities. A variety generated by a finite associative ring or by a finite Lie ring is a Specht variety. The question whether every variety of associative algebras is a Specht variety forms the content of the Specht problem. If a variety $\mathfrak M$ is generated by an associative algebra with a finite number of generators over a field of characteristic zero and if $M_2\not\subseteq\mathfrak M$, then $\mathfrak M$ is a Specht variety.

See also [[PI-algebra|PI-algebra]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  C. Procesi,   "Rings with polynomial identities" , M. Dekker  (1973)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" , Reidel  (1981)</TD></TR>
</table>



====Comments====
A.R. Kemer has shown that every variety of associative algebras over a field of characteristic zero is finitely based [[#References|[a1]]].

In addition to solving the Specht problem in the case of characteristic zero, Kemer has also described the $T$-ideal of a variety in terms of a finite $\mathbf Z_2$-graded algebra $A$ and an infinite Grassmannian algebra $G$. He has proved that, given a $T$-ideal $I$, there exists a finite-dimensional $\mathbf Z_2$-graded algebra $A=A_0\oplus A_1$ such that $I=T(A_0\otimes G_0+A_1\otimes G_1)$, the $T$-ideal of identities of the algebra $A_0\oplus G_0\otimes G_1$, where $G_0,G_1$ are the even and odd terms of the infinite Grassmannian algebra $G$.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.R. Kremer,   "Solution of the problem as to whether associative algebras have a finite basis of identities"  ''Soviet Math. Doklady'' , '''37'''  (1988)  pp. 60–64  ''Dokl. Akad. Nauk SSSR'' , '''298'''  (1988)  pp. 273–277</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  L.H. Rowen,   "Polynomial identities in ring theory" , Acad. Press  (1980)  pp. Chapt. 7</TD></TR>
</table>

[[Category:Associative rings and algebras]]