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

$I={\displaystyle \bigcap _{I_k\in S}}{I}_k$

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 $X$:

$(X)=\{{\displaystyle \sum _{\lambda }}(r_{\lambda }{a}_{\lambda }+n_{\lambda }{a}_{\lambda })\mid a_{\lambda }\in X,r_{\lambda }\in R,n_{\lambda }\in \mathbb{Z}\}.$

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