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

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

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

The notation $J(a)=S^{1}a{S}^1$ 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