For example, is a polynomial ring of one variable over A.

The next property of the polynomial ring is much deeper.

Matrices over a polynomial ring are important in the study of control theory.

Let be a polynomial ring over the finite field .

Note that unlike in an actual polynomial ring, the variables do not commute.

Let T be a regular chain in the polynomial ring R.

The substitution is a special case of the universal property of a polynomial ring.

Convert to a collection of polynomials in a certain polynomial ring with binary coefficients.

In a polynomial ring, it refers to its standard basis given by the monomials, .

The ring of power series can be seen as the completion of the polynomial ring.

Skew polynomial rings are closely related to crossed product algebras.

It states that every finitely generated projective module over a polynomial ring is free.

By the normalization lemma, A is integral over the polynomial ring .

The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators.

A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained.

A polynomial ring with coefficients in is the free commutative ring over its set of variables.

We denote by the corresponding polynomial ring.

The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree.

The grading on the polynomial ring is defined by letting each have degree one and every element of A, degree zero.

For this, they introduced the more general class of ideal lattices, which correspond to ideals in polynomial rings .

On top of this may be attached any number of symbolic variables thereby creating the polynomial ring and its quotient field.

In computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field.

Polynomial rings and their quotients by homogeneous ideals are typical graded algebras.

This ring is called polynomial ring.

For example, the invariants of group number 4 form a polynomial ring with 2 generators of degrees 4 and 6.