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ideal (Definition)

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.



"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

Attachments:
sum of ideals (Definition) by pahio
internal direct sum of ideals (Theorem) by Mathprof
regular ideal (Definition) by pahio
lattice of ideals (Definition) by CWoo
ideal of elements with finite order (Theorem) by pahio
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Cross-references: Dedekind domains, solutions, number fields, number theory, equivalent, commutative ring, subset, ring
There are 168 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 16087 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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