Ideal generated by a subset

Given a subset $A\subseteq R$ of a ring $R$ , the two-sided ideal generated by $A$ , denoted $(A)$ , is the smallest two-sided ideal of $R$ containing $A$ .

Generators in a unital ring

If $R$ is unital, $(A)$ consists of all finite sums

\sum_{i=1}^n r_i a_i s_i, \qquad a_i\in A,\quad r_i,s_i\in R,\quad n\ge0.

The empty sum is zero. If $R$ is also commutative, this reduces to finite sums $\sum_i r_i a_i$ . The smallest-ideal definition above also applies when $R$ is nonunital.

Remarks

When $A=\{a\}$ is a singleton and $R$ is commutative, $(A)$ is a principal ideal.

Examples

In $\mathbb Z$ , the ideal generated by $\{6,10\}$ is $(2)$ .
In $k[x,y]$ , the ideal generated by $\{x,y\}$ is $(x,y)$ .
In $M_2(k)$ , the two-sided ideal generated by a nonzero matrix is all of $M_2(k)$ .