# ideal

Let $S$ be a semigroup. An ideal of $S$ is a non-empty subset of $S$ which is closed under multiplication on either side by elements of $S$. Formally, $I$ is an ideal of $S$ if $I$ is non-empty, and for all $x\in I$ and $s\in S$, we have $sx\in I$ and $xs\in I$.

One-sided ideals are defined similarly. A non-empty subset $A$ of $S$ is a left ideal (resp. right ideal) of $S$ if for all $a\in A$ and $s\in S$, we have $sa\in A$ (resp. $as\in A$).

A principal left ideal of $S$ is a left ideal generated by a single element. If $a\in S$, then the principal left ideal of $S$ generated by $a$ is $S^{1}a=Sa\cup\{a\}$. (The notation $S^{1}$ is explained here (http://planetmath.org/AdjoiningAnIdentityToASemigroup3).)

Similarly, the principal right ideal generated by $a$ is $aS^{1}=aS\cup\{a\}$.

The notation $L(a)$ and $R(a)$ are also common for the principal left and right ideals generated by $a$ respectively.

A principal ideal of $S$ is an ideal generated by a single element. The ideal generated by $a$ is

 $S^{1}aS^{1}=SaS\cup Sa\cup aS\cup\{a\}.$

The notation $J(a)=S^{1}aS^{1}$ is also common.

 Title ideal Canonical name Ideal1 Date of creation 2013-03-22 13:05:43 Last modified on 2013-03-22 13:05:43 Owner mclase (549) Last modified by mclase (549) Numerical id 8 Author mclase (549) Entry type Definition Classification msc 20M12 Classification msc 20M10 Related topic ReesFactor Defines left ideal Defines right ideal Defines principal ideal Defines principal left ideal Defines principal right ideal