subring

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.

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.

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 never subrings since if they contain the identity then they must be the entire ring. For example, ideals in Z are of the form nZ where n is any integer. These are subrings if and only if n = ±1 (otherwise they do not contain 1) in which case they are all of Z.

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:

• The quaternion ring H contains only the complex plane as a planar subring
• The coquaternion ring contains three types of commutative planar subrings: the dual number plane, the split-complex number plane, as well as the ordinary complex plane
• The ring of 3 x 3 real matrices also contains 3-dimensional commutative subrings generated by the identity matrix and a nilpotent ε of order 3 (εεε = 0 ≠ εε). For instance, the Heisenberg group can be realized as the join of the groups of units of two of these nilpotent-generated subrings of 3 x 3 matrices.

References

• Iain T. Adamson (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3. 
• Page 84 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0 
• David Sharpe (1987). Rings and factorization. Cambridge University Press. pp. 15–17. ISBN 0-521-33718-6.