ideal generated by a subset of a ringIdealGeneratedByASet2004-09-28 12:31:462004-09-29 23:52:47Definitionideal generated byleft ideal generated byright ideal generated bygenerate as an idealgenerates as an idealgenerates% this is the default PlanetMath preamble. as your knowledge
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\renewcommand{\>}{\rangle}Let $X$ be a subset of a ring $R$. Let $S=\{I_k\}$ be the collection of all left ideals of $R$ that contain $X$ (note that the set is nonempty since $X\subset R$ and $R$ is an ideal in itself). The intersection
\begin{align*}
I=\bigcap_{I_k\in S} I_k
\end{align*}
is called the \emph{left ideal generated by $X$}, and is denoted by $(X)$. We say that $X$ \emph{generates} $I$ as an ideal.
The definition is symmetrical for right ideals.
Alternatively, we can constructively form the set of elements that constitutes this ideal: The left ideal $(X)$ consists of finite $R$-linear combinations of elements of $X$:
\begin{align*}
(X)=\left\{\sum_\lambda (r_\lambda a_\lambda + n_\lambda a_\lambda)\mid a_\lambda\in X, r_\lambda\in R, n_\lambda\in\Z\right\}.
\end{align*}