# ideal generated by a subset of a ring

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

 $\displaystyle I=\bigcap_{I_{k}\in S}I_{k}$

is called the left ideal generated by $X$, and is denoted by $(X)$. We say that $X$ 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 $X$:

 $\displaystyle(X)=\left\{\sum_{\lambda}(r_{\lambda}a_{\lambda}+n_{\lambda}a_{% \lambda})\mid a_{\lambda}\in X,r_{\lambda}\in R,n_{\lambda}\in\mathbb{Z}\right\}.$
 Title ideal generated by a subset of a ring Canonical name IdealGeneratedByASubsetOfARing Date of creation 2013-03-22 14:39:04 Last modified on 2013-03-22 14:39:04 Owner mathcam (2727) Last modified by mathcam (2727) Numerical id 9 Author mathcam (2727) Entry type Definition Classification msc 16D25 Related topic GeneratorsOfInverseIdeal Related topic PrimeIdealsByKrullArePrimeIdeals Defines ideal generated by Defines left ideal generated by Defines right ideal generated by Defines generate as an ideal Defines generates as an ideal Defines generates