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Let $R$ be a ring. A left ideal (resp., right ideal) $I$ of $R$ is a nonempty subset $I \subset R$ such that:
- $a-b \in I$ for all $a,b \in I$
- $r \cdot a \in I$ (resp. $a \cdot r \in I$ for all $a \in I$ and $r \in R$
A two-sided ideal is a left ideal $I$ which is also a right ideal. If $R$ is a commutative ring, then these three notions of ideal are equivalent. Usually, the word ``ideal'' by itself means two-sided ideal.
The name ``ideal'' comes from the study of number theory. When the failure of unique factorization in number fields was first noticed, one of the solutions was to work with so-called ``ideal numbers'' in which unique factorization did hold. These ``ideal numbers'' were in fact ideals, and in Dedekind domains, unique factorization of ideals does indeed hold. The term ``ideal number'' is no longer used; the term ``ideal'' has replaced and generalized it.
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"ideal" is owned by djao. [ full author list (2) ]
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See Also: subring, prime ideal
| Also defines: |
left ideal, right ideal, 2-sided ideal, two-sided ideal |
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Cross-references: Dedekind domains, solutions, number fields, unique factorization, number theory, equivalent, commutative ring, subset, ring
There are 178 references to this entry.
This is version 13 of ideal, born on 2001-10-19, modified 2007-12-29.
Object id is 371, canonical name is Ideal.
Accessed 19373 times total.
Classification:
| AMS MSC: | 13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory) | | | 11N80 (Number theory :: Multiplicative number theory :: Generalized primes and integers) | | | 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals) | | | 14K99 (Algebraic geometry :: Abelian varieties and schemes :: Miscellaneous) |
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Pending Errata and Addenda
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