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ideal generated by a subset of a ring (Definition)

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$:

$\displaystyle (X)=\left\{\sum_\lambda (r_\lambda a_\lambda + n_\lambda a_\lambda)\mid a_\lambda\in X, r_\lambda\in R, n_\lambda\in\mathbb{Z}\right\}.$    



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See Also: generators of inverse ideal

Also defines:  ideal generated by, left ideal generated by, right ideal generated by, generate as an ideal, generates as an ideal, generates

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entries on finitely generated ideals (Topic) by pahio
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Cross-references: combinations, finite, right ideals, intersection, ideal, contain, left ideals, collection, ring, subset
There are 60 references to this entry.

This is version 6 of ideal generated by a subset of a ring, born on 2004-09-28, modified 2004-09-29.
Object id is 6242, canonical name is IdealGeneratedByASet.
Accessed 8999 times total.

Classification:
AMS MSC: 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)
Pending Errata and Addenda
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