subringe

Subringe is a term occasionally used to denote a subring of a ring. The standard term in most modern algebra texts is subring, and many authors treat “subringe” as a historical or regional variant. In any case, the concept aligns with the usual definition of a subring: a subset that inherits ring operations from a parent ring.

Definition: Let R be a ring (not necessarily with unity). A subset S ⊆ R is a subringe of R if S is closed under addition and multiplication, and contains the zero element of R. Depending on convention, a subringe may or may not be required to contain the multiplicative identity 1R; when 1R ∈ S, S is called a unital subringe.

Examples: In the ring Z of integers, the set nZ for any integer n is a subring (and a unital subringe only for n = ±1). The set of even integers is a subringe of Z. In the polynomial ring R[x], the set of polynomials with zero constant term forms a subringe.

Properties: The intersection of any family of subrings (subrings) is a subringe. The subringe generated by a subset A ⊆ R is the smallest subringe containing A. Subringe structure inherits many properties from the ambient ring, for instance it remains closed under subtraction, since subtraction is addition of additive inverses.

Relation to subfields: If R is a field, any subringe of R that contains 1R is a subring and is an integral domain; it is a subfield precisely when every nonzero element has a multiplicative inverse in the subringe.

See also: Subring, Ring theory, Integral domain, Field.