Subring

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In mathematics, a subring is a subset of a ring which contains the multiplicative identity and is itself a ring under the same binary operations. Naturally, those authors who do not require rings to contain a multiplicative identity do not require subrings to possess the identity (if it exists). This leads to the added advantage that ideals become subrings (see below).

A subring of a ring (R, +, *) is a subgroup of (R, +) which contains the mutiplicative identity and is closed under multiplication.

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 ring Z has no subrings (with multiplicative identity) other than itself, and the zero ring.

Every ring has a unique smallest subring, isomorphic to either the integers Z or some ring Z/nZ with n a nonnegative integer (see characteristic).

The subring test states that for any ring, a nonempty subset of that ring is itself a ring if it is closed under multiplication and subtraction, and has a multiplicative identity.

Contents

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 that are closed under both left and right multiplication by elements from R.

If one omits the requirement that rings have a unit element, then subrings need only be non-empty and be closed under subtraction 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 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 by the variety of commutative subrings that it hosts:

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