# T-ideal

Let $R$ be a commutative ring and $R\u27e8X\u27e9$ be a free algebra^{} over $R$ on a set $X$ of *non-commuting* variables. A two-sided ideal^{} $I$ of $R\u27e8X\u27e9$ is called a $T$-*ideal* if $\varphi (I)\subseteq I$ for any $R$-endomorphism^{} $\varphi $ of $R\u27e8X\u27e9$.

For example, let $A$ be a $R$-algebra^{}. Define $\mathcal{T}(A)$ to be the set of all polynomial identities (http://planetmath.org/PolynomialIdentityAlgebra) $f\in R\u27e8X\u27e9$ for $A$. Then $\mathcal{T}(A)$ is a $T$-ideal of $R\u27e8X\u27e9$. $\mathcal{T}(A)$ is called the $T$-*ideal of of A*.

Title | T-ideal |
---|---|

Canonical name | Tideal |

Date of creation | 2013-03-22 14:21:12 |

Last modified on | 2013-03-22 14:21:12 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 7 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 16R10 |