Definition:Generated Ideal of Ring
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Definition
Let $\struct {R, +, \circ}$ be a ring.
Let $S \subseteq R$ be a subset.
Definition 1
The ideal generated by $S$ is the smallest ideal of $R$ containing $S$, that is, the intersection of all ideals containing $S$.
Definition 2: for commutative rings with unity
Let $R$ be a commutative ring with unity.
The ideal generated by $S$ is the set of all linear combinations of elements of $S$.
Definition 3: for rings with unity
Let $R$ be a ring with unity.
The ideal generated by $S$ is the set of all two-sided linear combinations of elements of $S$.
Also see
- Equivalence of Definitions of Generated Ideal of Ring
- Definition:Generator of Ideal
- Definition:Generated Subring
- Generated Ideal of Ring is Closure Operator