Variety of rings

2020 Mathematics Subject Classification: Primary: 08B [MSN][ZBL]

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). Since the totality of polynomial identities that are satisfied in a given ring forms a fully-characteristic ideal (a $T$-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.

References

[1]	C. Procesi, "Rings with polynomial identities" , M. Dekker (1973)
[2]	P.M. Cohn, "Universal algebra" , Reidel (1981)

Comments

A.R. Kemer has shown that every variety of associative algebras over a field of characteristic zero is finitely based [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

[a1]	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
[a2]	L.H. Rowen, "Polynomial identities in ring theory" , Acad. Press (1980) pp. Chapt. 7

Comments

A Schreier variety $\mathfrak{S}$ has the property that every subalgebra of an $\mathfrak{S}$-free algebra is $\mathfrak{S}$-free. Examples of Schreier varieties in the context of algebras over a field include the variety of non-associative algebras; non-associative commutative algebras; non-associative anti-commutative algebras (cf. Non-associative rings and algebras); associative algebras (cf. Associative rings and algebras; associative commutative algebras; Lie algebras; Lie $p$-algebras; Lie superalgebras. The variety of Leibnitz algebras is not a Schreier variety.

References

[b1]	Mikhalev, Alexander A.; Shpilrain, Vladimir; Yu, Jie-Tai, Combinatorial methods. Free groups, polynomials, and free algebras, CMS Books in Mathematics 19 Springer (2004) ISBN 0-387-40562-3 Zbl 1039.16024