Definition:Generated Ideal of Ring/Commutative and Unitary
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Definition
Let $\struct {R, +, \circ}$ be a commutative ring with unity.
Let $S \subseteq R$ be a subset of $R$.
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
Sources
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields