Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra In mathematics, a subring of a ring R is a subset of R that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and that shares the same multiplicative identity as R.[a] A subring of a ring (R, +, *, 0, 1) is a subset S of R that preserves the structure of the ring, i.e. a ring (S, +, *, 0, 1) with S ⊆ R. Equivalently, it is both a subgroup of (R, +, 0) and a submonoid of (R, *, 1).
In this case, a subring of R is a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R).
This alternate definition gives a strictly weaker condition, even for rings that do have a multiplicative identity, in that all ideals become subrings, and they may have a multiplicative identity that differs from the one of R. With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of R that is a subring of R is R itself.
[3] The subring generated by X is also the set of all linear combinations with integer coefficients of elements of X, including the additive identity ("empty combination") and multiplicative identity ("empty product").
Individual elements can also be adjoined to a subring, denoted R[a1, a2, ..., an].