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 . It is easy to verify that is a homomorphism. Suppose that is a homomorphism with , . Then by direct computation. Define a category whose objects are tuples where is a commutative ring with identity, , a homomorphism with . A morphism in from to is a homomorphism such that , and . Verify that is an equivalence in iff is an equivalence of rings. If