Definition:Generated Ideal of Ring

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Definition

Let $\struct {R, +, \circ}$ be a ring with unity.

Let $S \subseteq R$ be a subset of $R$.


The ideal generated by $S$ is the smallest ideal of $R$ containing $S$, that is, the intersection of all ideals containing $S$.


Unitary Rings

The ideal generated by $S$ is the set of all two-sided linear combinations of elements of $S$.


Commutative and Unitary Rings

The ideal generated by $S$ is the set of all linear combinations of elements of $S$.


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


Generalizations


Sources

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