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ideal generated by a subset of a ring
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(Definition)
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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
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 $X$ :
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Cross-references: combinations, finite, right ideals, intersection, ideal, contain, left ideals, collection, ring, subset
There are 56 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 11929 times total.
Classification:
| AMS MSC: | 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals) |
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Pending Errata and Addenda
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