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ideal (Definition)

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^1a = Sa \cup \{a\}$. (The notation $ S^1$ is explained here.)

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

$\displaystyle S^1aS^1 = SaS \cup Sa \cup aS \cup \{a\}.$
The notation $ J(a) = S^1aS^1$ is also common.



"ideal" is owned by mclase.
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See Also: Rees factor

Also defines:  left ideal, right ideal, principal ideal, principal left ideal, principal right ideal
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Cross-references: ideal generated by, generated by, left ideal generated by, side, multiplication, closed under, subset, semigroup
There are 25 references to this entry.

This is version 5 of ideal, born on 2002-10-10, modified 2003-08-29.
Object id is 3516, canonical name is Ideal3.
Accessed 9587 times total.

Classification:
AMS MSC: 20M12 (Group theory and generalizations :: Semigroups :: Ideal theory)
 20M10 (Group theory and generalizations :: Semigroups :: General structure theory)
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