ideal Ideal 2001-10-19 01:28:08 2007-12-29 23:48:54 Definition left ideal right ideal 2-sided ideal two-sided ideal \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} \PMlinkescapeword{term} Let $R$ be a ring. A \emph{left ideal} (resp., \emph{right ideal}) $I$ of $R$ is a nonempty subset $I \subset R$ such that: \begin{itemize} \item $a-b \in I$ for all $a,b \in I$ \item $r \cdot a \in I$ (resp. $a \cdot r \in I$) for all $a \in I$ and $r \in R$ \end{itemize} A \emph{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.