Definition:Generated Ideal of Ring

Definition

Let $\left({R, +, \circ}\right)$ 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 $S$.