Polynomial rings

The ring is called the ring of polynomials over .

If has an identity, and we write polynomials as . If is a ring without identity, embed to a ring with identity. Identify with its image under the embedding map so that is a subring of . Then is a subring of . Thus every polynomial can be written uniquely as where , . If , the elements are called the coefficients of . The element is called the constant term. Elements of which have the form are called constant polynomials. If and , then is called the leading coefficient of . If has an identity and has leading coefficient , then is said to be a monic polynomial.

Let be a ring with identity. The element of is called an indeterminate. If is another ring with identity, the indeterminate is not the same element as . We can also define polynomials in more than one indeterminate.

The ring is called the ring of polynomials in determinates over . is considered a subring of . Let be a positive integer and for , let where is the th coordinate of . Every element of may be written as .

If is a ring with identity, the elements are called indeterminates. The elements are called coefficients of the polynomial . A polynomial of the form is called a monomial. The notation and terminology is extended to polynomial rings where has no identity. Embed the ring to a ring with identity and consider as a subring of . If is any ring, for any subset , the monomorphism exists.

Let be a homomorphism of rings, , . Let . Let

Proof. If , then . The map given by is a well-defined map such that and