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Milds # Subring

In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. (Note that a subset of a ring R need not be a ring.) For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of R). With definition requiring a multiplicative identity (which is used in this article), the only ideal of R that is a subring of R is R itself.

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• Ring Theory | Subring | Theorems & Examples Of Subring | Abstract Algebra
• Ring Definition (expanded) - Abstract Algebra
• Ring Examples (Abstract Algebra)
• Ring | Subring | Discrete mathematics
• Abstract Algebra | 12. Subrings

## Definition

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 SR. Equivalently, it is both a subgroup of (R, +, 0) and a submonoid of (R, ∗, 1).

## Examples

The ring $\mathbb {Z}$ and its quotients $\mathbb {Z} /n\mathbb {Z}$ have no subrings (with multiplicative identity) other than the full ring.: 228

Every ring has a unique smallest subring, isomorphic to some ring $\mathbb {Z} /n\mathbb {Z}$ with n a nonnegative integer (see Characteristic). The integers $\mathbb {Z}$ correspond to n = 0 in this statement, since $\mathbb {Z}$ is isomorphic to $\mathbb {Z} /0\mathbb {Z}$ .: 89–90

## Subring test

The subring test is a theorem that states that for any ring R, a subset S of R is a subring if and only if it is nonempty and closed under multiplication and subtraction.: 228

As an 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].

## Ring extensions

If S is a subring of a ring R, then equivalently R is said to be a ring extension of S, written as R/S in similar notation to that for field extensions.

## Subring generated by a set

Let R be a ring. Any intersection of subrings of R is again a subring of R. Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R. S is the smallest subring of R containing X. ("Smallest" means that if T is any other subring of R containing X, then S is contained in T.) S is said to be the subring of R generated by X. If S = R, we may say that the ring R is generated by X.

## Relation to ideals

Proper ideals are subrings (without unity) that are closed under both left and right multiplication by elements of R.

If one omits the requirement that rings have a unity element, then subrings need only be non-empty and otherwise conform to the ring structure, 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 unity, and a "subring-without-unity", but not a "subring-with-unity" of Z × Z.
• The proper ideals of Z have no multiplicative identity.

## Profile by commutative subrings

A ring may be profiled[clarification needed] by the variety of commutative subrings that it hosts: