In mathematics, a subring of a ring R is a subset of R 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 R.[a]
A subring of a ring (R, +, *, 0, 1) is a subset S of R that preserves the structure of the ring, i.e. a ring (S, +, *, 0, 1) with S ⊆ R. Equivalently, it is both a subgroup of (R, +, 0) and a submonoid of (R, *, 1).
Equivalently, S is a subring if and only if it contains the multiplicative identity of R, and is closed under multiplication and subtraction. This is sometimes known as the subring test.[1]
Some mathematicians define rings without requiring the existence of a multiplicative identity (see Ring (mathematics) § History). In this case, a subring of R is a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). 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 R. With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of R that is a subring of R is R itself.
A special kind of subring of a ring R is the subring generated by a subset X, which is defined as the intersection of all subrings of R containing X.[3] The subring generated by X is also the set of all linear combinations with integer coefficients of elements of X, including the additive identity ("empty combination") and multiplicative identity ("empty product").[citation needed]
Any intersection of subrings of R is itself a subring of R; therefore, the subring generated by X (denoted here as S) is indeed a subring of R. This subring S is the smallest subring of R containing X; that is, if T is any other subring of R containing X, then S ⊆ T.
Since R itself is a subring of R, if R is generated by X, it is said that the ring R is generated by X.
Subrings generalize some aspects of field extensions. If S is a subring of a ring R, then equivalently R is said to be a ring extension[b] of S.
If A is a ring and T is a subring of A generated by R ∪ S, where R is a subring, then T is a ring extension and is said to be S adjoined to R, denoted R[S]. Individual elements can also be adjoined to a subring, denoted R[a1, a2, ..., an].[4][3]
For example, the ring of Gaussian integers is a subring of generated by , and thus is the adjunction of the imaginary unit i to .[3]
The intersection of all subrings of a ring R is a subring that may be called the prime subring of R by analogy with prime fields.
The prime subring of a ring R is a subring of the center of R, which is isomorphic either to the ring of the integers or to the ring of the integers modulo n, where n is the smallest positive integer such that the sum of n copies of 1 equals 0.
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