In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ...

More precisely, given a ring (R, +, *), we say that a subset S of R is a subring of R if it is a ring under the restriction of + and * to S, and contains the same multiplicitive identity as R. A subring is just a subgroup of (R, +) which contains the identity and is closed under multiplication. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...

For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[X]. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

The ring Z has no subrings other than itself. Note that ideals in Z, which are of the form nZ, where n is any integer, are not subrings (unless n = ±1) as they do not contain 1. In general, a proper ideal is never a subring since if it contains the identity then it must be the entire ring. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

If one omits the requirement that rings have a unit element, then subrings need only be closed under addition and multiplication and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring):

• The ideal I = {(z,0)|z in Z} of the ring Z × Z = {(x,y)|x,y in Z} with componentwise addition and multiplication has the identity (1,0), which is different from the identity (1,1) of the ring. So I is a ring with unit, and a "subring-without-unit", but not a "subring-with-unit" of Z × Z.
• The proper ideals of Z have no multiplicative identity.

Every ring has a unique smallest subring, isomorphic to either the integers Z or some cyclic group Z/nZ (see characteristic). In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...