# T-ideal

Let $R$ be a commutative ring and $R\langle X\rangle$ be a free algebra over $R$ on a set $X$ of non-commuting variables. A two-sided ideal $I$ of $R\langle X\rangle$ is called a $T$-ideal if $\phi(I)\subseteq I$ for any $R$-endomorphism $\phi$ of $R\langle X\rangle$.

For example, let $A$ be a $R$-algebra. Define $\mathcal{T}(A)$ to be the set of all polynomial identities (http://planetmath.org/PolynomialIdentityAlgebra) $f\in R\langle X\rangle$ for $A$. Then $\mathcal{T}(A)$ is a $T$-ideal of $R\langle X\rangle$. $\mathcal{T}(A)$ is called the $T$-ideal of of A.

Title T-ideal Tideal 2013-03-22 14:21:12 2013-03-22 14:21:12 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 16R10