# T-ideal

of a free associative algebra

A totally invariant ideal, that is, an ideal invariant under all endomorphisms. The set of all polynomial identities of an arbitrary variety of associative algebras over a field $F$ (cf. Associative rings and algebras) forms a $T$-ideal in the countably-generated free algebra $F[X]$, $X=\{x_1,\dots,x_k,\dots\}$. Thus, there exists a one-to-one correspondence between the $T$-ideals of $F[X]$ and the varieties of associative algebras over $F$. If $F$ has characteristic $0$, then for every $T$-ideal $T\subseteq F[X]$ there exists a natural number $n=n(T)$ such that certain powers of elements of $M_n(F)$ are elements of $T$, and only they, where $M_n(F)$ is the ideal of identities of the algebra $F_n$ of all $(n\times n)$-matrices over $F$. In this case a $T$-ideal can also be defined as a (two-sided) ideal that is closed under all differentiations of the free algebra. The quotient algebra $F[X]/T$ is a PI-algebra with $T$ as set of polynomial identities. It is called the relatively free algebra (or generic algebra) with $T$-ideal of identities $T$ (and is a free algebra in the variety of algebras defined by the identities in $T$). The algebra $F[X]/T$ has no zero divisors if and only if $T=M_n(F)$ for some natural number $n$. Every $T$-ideal $T$ of a free associative algebra is primary.

The $T$-ideals of a free associative algebra on infinitely many generators over a field of characteristic zero form a free semi-group under the operation of multiplication of ideals. In this case a $T$-ideal can be defined as an ideal invariant under all automorphisms of the free algebra.

For the question as to whether every $T$-ideal of $F[X]$ is the totally invariant closure of finitely many elements (Specht's problem) see also Variety of rings.

$T$-ideals can be defined for non-associative algebras (Lie, alternative and others) by analogy with the associative case.

How to Cite This Entry:
T-ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=T-ideal&oldid=33191
This article was adapted from an original article by V.N. Latyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article