ideal

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.

 Title ideal Canonical name Ideal Date of creation 2013-03-22 11:49:27 Last modified on 2013-03-22 11:49:27 Owner djao (24) Last modified by djao (24) Numerical id 18 Author djao (24) Entry type Definition Classification msc 14K99 Classification msc 16D25 Classification msc 11N80 Classification msc 13A15 Classification msc 54C05 Classification msc 54C08 Classification msc 54J05 Classification msc 54D99 Classification msc 54E05 Classification msc 54E15 Classification msc 54E17 Related topic Subring Related topic PrimeIdeal Defines left ideal Defines right ideal Defines 2-sided ideal Defines two-sided ideal