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, SubringThis entry contributed by Margherita Barile
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Barile, Margherita. "Extension Ring." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ExtensionRing.html