Extension Ring

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 in S, the set

 R[a]={f(a)|f(x) in 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) in R[x_1,...,x_n]},

which is the ring extension of R in S generated by a_1,...,a_n.

See also

Extension Field, Group Extension, Integral Extension, Integral Closure, Integrally Closed, Proper Extension, Ring of Fractions, Subring

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Extension Ring." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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