Definition:Generator of Ideal of Ring
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Definition
Let $R$ be a commutative ring.
Let $I \subset R$ be an ideal.
Let $S \subset I$ be a subset.
Then:
- $S$ is a generator of $I$
- $I$ is the ideal generated by $S$.
Notation
For a ring $R$, let $S \subseteq R$ be a generator of an ideal $\II$ of $R$.
Then we write:
- $\II = \gen S$
If $S$ is a singleton, that is: $S = \set x$, then we can (and usually do) write:
- $\II = \gen x$
for the ideal generated by $\set x$, rather than:
- $\II = \gen {\set x}$
Where $\map P x$ is a propositional function, the notation:
- $\II = \gen {x \in S: \map P x}$
can be seen for:
- $\II = \gen {\set {x \in S: \map P x} }$
which is no more than notation of convenience.
Also see
- Results about generators of ideals can be found here.