# 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