Ideal generated by a subset

Given a subset $A\subseteq R$ of a ring $R$, the ideal generated by $A$, denoted $(A)$, is the smallest ideal of $R$ containing $A$. Equivalently, $(A)$ consists of all finite sums of the form

with $n\ge 1$, $a_i\in A$, and $r_i,s_i\in R$ (in the commutative case, one may take $s_i=1$).

When $A=\{a\}$ is a singleton, $(A)$ is a principal ideal . This construction is functorial in the sense that homomorphisms send generated ideals to generated ideals of images (up to containment).

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 $M_2(k)$.