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A polynomial ring is a fundamental algebraic structure in abstract algebra. It consists of polynomials in one or more indeterminates with coefficients taken from a given ring, R. The set of polynomials forms a ring under the usual operations of polynomial addition and multiplication.

The simplest examples are polynomial rings in a single indeterminate, denoted as R[x]. For instance, the ring of polynomials in x with integer coefficients is denoted by Z[x]. Elements of Z[x] are expressions like 3x^2 - 5x + 1. The addition and multiplication of these polynomials follow standard algebraic rules.

When considering polynomials in multiple indeterminates, such as x and y, the notation extends to R[x, y]. An element in R[x, y] might look like 2x^2y - 3xy^3 + 7. The ordering of terms in multivariate polynomials can be defined using various conventions, such as lexicographical order or graded lexicographical order, which are important for algorithms and theoretical properties.

Polynomial rings possess many rich properties. For example, if R is an integral domain, then R[x] is also an integral domain. The concept of unique factorization domains (UFDs) and principal ideal domains (PIDs) is closely related to polynomial rings. For instance, if F is a field, then the polynomial ring F[x] is a Euclidean domain and thus a PID and a UFD.

These structures are crucial in various fields of mathematics, including algebraic geometry, number theory, and coding theory, serving as building blocks for more complex mathematical objects and theories.