Definition:Generated Ideal of Ring

Commutative Ring
Let $\left({R, +, \circ}\right)$ be a commutative 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
Let $R$ be a commutative ring with unity.

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

Noncommutative ring
Let $\left({R, +, \circ}\right)$ be a ring.

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
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

Generalizations

 * Definition:Generated Submodule