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/"n"Z 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.

ubring 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 "n"Z 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 contain 0 and be closed under addition, 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

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