In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and that shares the same multiplicative identity as .In general, not all subsets of a ring are rings.
Definition A subring of a ring is a subset of that preserves the structure of the ring, i.e. a ring with . Equivalently, it is both a subgroup of and a submonoid of .
Equivalently, is a subring if and only if it contains the multiplicative identity of , and is closed under multiplication and subtraction. This is sometimes known as the subring test.
Variations Some mathematicians define rings without requiring the existence of a multiplicative identity (see ). In this case, a subring of is a subset of that is a ring for the operations of (this does imply it contains the additive identity of ). This alternate definition gives a strictly weaker condition, even for rings that do have a multiplicative identity, in that all ideals become subrings, and they may have a multiplicative identity that differs from the one of . With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of that is a subring of is itself.
Examples The ring of integers is a subring of both the field of real numbers and the polynomial ring .
and its quotients have no subrings (with multiplicative identity) other than the full ring.
Every ring has a unique smallest subring, isomorphic to some ring with n a nonnegative integer (see Characteristic). The integers correspond to in this statement, since is isomorphic to .
The center of a ring is a subring of , and is an associative algebra over its center.
Subring generated by a set
A special kind of subring of a ring is the subring generated by a subset , which is defined as the intersection of all subrings of containing . The subring generated by is also the set of all linear combinations with integer coefficients of products of elements of , including the additive identity ("empty combination") and multiplicative identity ("empty product").
Any intersection of subrings of is itself a subring of ; therefore, the subring generated by (denoted here as ) is indeed a subring of . This subring is the smallest subring of containing ; that is, if is any other subring of containing , then .
Since itself is a subring of , if is generated by , it is said that the ring is generated by .
Ring extension Subrings generalize some aspects of field extensions. If is a subring of a ring , then equivalently is said to be a ring extensionNot to be confused with the ring-theoretic analog of a group extension. of .
Adjoining If is a ring and is a subring of generated by , where is a subring, then is a ring extension and is said to be adjoined to , denoted . Individual elements can also be adjoined to a subring, denoted .
For example, the ring of Gaussian integers is a subring of generated by , and thus is the adjunction of the imaginary unit to .
Prime subring The intersection of all subrings of a ring is a subring that may be called the prime subring of by analogy with prime fields.
The prime subring of a ring is a subring of the center of , which is isomorphic either to the ring of the integers or to the ring of the integers modulo , where is the smallest positive integer such that the sum of copies of equals .
See also Integral extension Group extension Algebraic extension Ore extension
Notes
References
General references