subrings

A subring of a ring R is a subset S of R that is itself a ring under the same addition and multiplication as R. Concretely, S must be closed under addition and multiplication, contain the additive identity 0, and be closed under additive inverses (so for any s in S, −s is in S). In many texts, a subring is allowed to lack the multiplicative identity of R; in others, a subring is required to contain 1_R and is then called a unital or subring with unity. The convention varies by author.

Key properties include that the intersection of any family of subrings of R is again a subring, and that a subring inherits all ring-theoretic properties of the parent ring in a restricted form. A subring is not required to be an ideal; ideals are special subrings that are additionally closed under multiplication by arbitrary elements of R.

Examples: The set of integers Z is a subring of the rational numbers Q, the real numbers R, and the complex numbers C. For any ring R, the multiples nZ form a subring of Z. In the ring of n by n matrices over a field F, the diagonal matrices form a subring (and contain the identity matrix, making it a unital subring). Subrings can also be generated: the subring generated by a subset A of R is the smallest subring containing A, often described as finite sums of finite products of elements of A and their additive inverses.

Relationships: Every ring homomorphism image is a subring of the codomain, and subrings relate to broader notions such as subalgebras when a ring carries additional structure.