T-ideal

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

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