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". 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.
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 ring formula_1 and its quotients formula_2 have no subrings (with multiplicative identity) other than the full ring.
Every ring has a unique smallest subring, isomorphic to some ring formula_2 with "n" a nonnegative integer (see characteristic). The integers formula_1 correspond to in this statement, since formula_1 is isomorphic to formula_6.
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 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):
If "I" is a prime ideal of a commutative ring "R", then the intersection of "I" with any subring "S" of "R" remains prime in "S". In this case one says that "I" lies over "I" ∩ "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: