idealIdeal32002-10-10 18:02:102003-08-29 12:18:29Definitionleft idealright idealprincipal idealprincipal left idealprincipal right ideal% this is the default PlanetMath preamble. as your knowledge
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Let $S$ be a semigroup. An \emph{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 \emph{left ideal} (resp. \emph{right ideal}) of $S$ if for all $a \in A$ and $s \in S$, we have $sa \in A$ (resp. $as \in A$).
A \emph{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 $S^1a = Sa \cup \{a\}$. (The notation $S^1$ is explained \PMlinkname{here}{AdjoiningAnIdentityToASemigroup3}.)
Similarly, the \emph{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 \emph{principal ideal} of $S$ is an ideal generated by a single element. The ideal generated by $a$ is $$S^1aS^1 = SaS \cup Sa \cup aS \cup \{a\}.$$ The notation $J(a) = S^1aS^1$ is also common.