ring extension

A extension ring (or ring extension) of a ring R is any ring S of which R is a subring. For example, the field of rational numbers Q and the ring of Gaussian integers Z[i] are extension rings of the ring of integers Z. For every ring R, the polynomial ring R[x] is a ring extension of R. If S is a ring extension of R, and a element S, the set R[a] = {f(a)|f(x) element R[x]}, is the smallest subring of S containing R and a, and is a ring extension of R. More generally, given finitely many elements a_1, ..., a_n of S, we can consider R[a_1, ..., a_n] = {f(a_1, ..., a_n)|f(x_1 ..., x_n) element R[x_1, ..., x_n]}, which is the ring extension of R in S generated by a_1, ..., a_n.

extension field | group extension | integral closure | integral extension | integrally closed | proper extension | ring of fractions | subring