Subring
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.
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. In this case, a subring of is a subset of that is a ring for the operations 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 other than the full ring.
- Every ring has a unique smallest subring, isomorphic to some ring with n a nonnegative integer. 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
Any intersection of subrings of is itself a subring of ; therefore, the subring generated by 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 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.