Definition:Generator of Ideal of Ring

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 $I$

if and only if:

$I$ 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

• Definition:Generated Ideal of Ring

• Results about generators of ideals can be found here.

Generalizations

• Definition:Generator of Module