ideal

《代数学引论》 · 第 50 页 · 定义 15

设 $L$ 是一环，$I\subset L$ 是 $L$ 的一个加法子群。如果对于任意 $r\in L$，$a\in I$ 都有 $ra\in I$，$ar\in I$，那么 $I$ 称为 $L$ 的一个理想（或称双边理想）。

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《Abstract Algebra》 · 第 242 页 · Section 7.3「Ring Homomorphisms and Quotient Rings」，Definition

Let $R$ be a ring, let $I$ be a subset of $R$ and let $r\in R$. (1) $rI=\{ra\mid a\in I\}$ and $Ir=\{ar\mid a\in I\}$. (2) A subset $I$ of $R$ is a left ideal of $R$ if (i) $I$ is a subring of $R$, and (ii) $I$ is closed under left multiplication by elements from $R$, i.e., $rI\subseteq I$ for all $r\in R$. Similarly $I$ is a right ideal if (i) holds and in place of (ii) one has (ii)' $I$ is closed under right multiplication by elements from $R$, i.e., $Ir\subseteq I$ for all $r\in R$. (3) A subset $I$ that is both a left ideal and a right ideal is called an ideal (or, for added emphasis, a two-sided ideal) of $R$.