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product of ideals (Definition)

Let $ R$ be a ring, and let $ A$ and $ B$ be left (right) ideals of $ R$. Then the product of the ideals $ A$ and $ B$, which we denote $ AB$, is the left (right) ideal generated by all products $ ab$ with $ a\in A$ and $ b\in B$. Note that since sums of products of the form $ ab$ with $ a\in A$ and $ b\in B$ are contained simultaneously in both $ A$ and $ B$, we have $ AB\subset A\cap B$.



"product of ideals" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: sum of ideals, quotient of ideals, Prüfer ring, product of left and right ideal


Attachments:
product of finitely generated ideals (Definition) by pahio
ideal multiplication laws (Definition) by pahio
product of left and right ideal (Theorem) by pahio
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Cross-references: contained, sums, ideal generated by, product, ideals, right, ring
There are 9 references to this entry.

This is version 6 of product of ideals, born on 2001-10-20, modified 2008-03-22.
Object id is 411, canonical name is ProductOfIdeals.
Accessed 4253 times total.

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