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. 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. The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings. With definition requiring a multiplicative identity, the only ideal of R that is a subring of R is R itself.
Definition
A subring of a ring is a subset S of R that preserves the structure of the ring, i.e. a ring with. Equivalently, it is both a subgroup of and a submonoid of.
Examples
The ringZ and its quotients Z/nZ have no subrings other than the full ring. Every ring has a unique smallest subring, isomorphic to some ring Z/nZ with n a nonnegative integer. The integers Z correspond to in this statement, since Z is isomorphic to Z/0Z.
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 closed under multiplication and subtraction, and contains the multiplicative identity of R. 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 polynomialsZ.
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.
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. S is said to be the subring of Rgenerated 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 unity element, then subrings need only be non-empty and otherwiseconform to the ring structure, and ideals become subrings. Ideals may or may not have their own multiplicative identity :
The ideal I = of the ring Z × Z = with componentwise addition and multiplication has the identity, which is different from the identity 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.
If I is a prime ideal of a commutative ringR, then the intersection of I with any subring S of R remains prime in S. In this case one says that Ilies overI ∩ S. The situation is more complicated when R is not commutative.
Profile by commutative subrings
A ring may be profiled by the variety of commutative subrings that it hosts:
