Subring

In ring theory, a subring of a ring R is a subset S ⊆ R that is a ring under the same operations as R. Equivalently, S is an additive subgroup of R closed under multiplication. Therefore, S contains the zero element and is closed under addition and subtraction and under multiplication. In rings with a multiplicative identity, some authors require 1_R ∈ S (making S a unital subring), while others allow subrings that do not contain 1_R (nonunital subrings).

Examples include: the integers Z as a subring of the rationals Q or the reals R; the even integers 2Z form a subring of Z, but they do not contain 1; the polynomial ring Z[x] is a subring of Q[x] (and of R[x]), and 2Z[x] is a nonunital subring of Z[x]; within matrix rings, the set of diagonal matrices with integer entries is a subring of M_n(Z); the set of polynomials in R[x] with zero constant term is a subring of R[x].

Basic properties: the intersection of any collection of subrings of R is a subring; a subring need not be an ideal, but every ideal is a subring (and in addition absorbs multiplication by arbitrary elements of R). Subrings inherit many structural features from R, such as commutativity in the case R is commutative, and they play a central role in constructing new rings and in studying ring extensions.