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# 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 $f\in R\langle X\rangle$ for $A$. Then $\mathcal{T}(A)$ is a $T$-ideal of $R\langle X\rangle$. $\mathcal{T}(A)$ is called the $T$-*ideal of identities of A*.

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