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.