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Extension Ring

An extension ring (or ring extension) of a ring is any ring of which is a subring. For example, the field of rational numbers and the ring of Gaussian integers are extension rings of the ring of integers .

In the following adjoining-element formulas, suppose is a commutative ring and is a subring with the same multiplicative identity.

For every ring , the polynomial ring is a ring extension of . If is a ring extension of , and , the set

is the smallest subring of containing and , and is a ring extension of . More generally, given finitely many elements of , 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 in generated by .

For an extension of unit rings with the same multiplicative identity, allowing noncommutative rings, polynomial evaluation gives a ring homomorphism fixing only when the substituted element commutes with every element of . With several indeterminates, the substituted elements must also commute with one another. Without these conditions, the generated subring consists of sums of finite words in the adjoined elements and coefficients from , rather than just ordinary polynomial expressions (Sharifi, §3.6).