Subring

A subring of a ring $R$ is a nonempty subset $S\subseteq R$ such that, with the operations induced from $R$, the triple $(S,+,\cdot)$ is a ring. Equivalently, $S$ is closed under addition, additive inverses, and multiplication (and hence contains $0$).

If $R$ is unital , one sometimes distinguishes unital subrings by additionally requiring $1_R\in S$; unless stated, “subring” need not contain the identity. Subrings often appear as images of homomorphisms and as ambient structures for ideal theory.

Examples:

• $\mathbb Z$ is a subring of $\mathbb Q$.
• The diagonal matrices form a subring of $M_n(\mathbb Z)$.
• $2\mathbb Z$ is a subring of $\mathbb Z$, but it is not a unital subring.