A polynomial ring is a mathematical structure formed by polynomials, which are expressions made up of variables and coefficients. In a polynomial ring, you can add, subtract, and multiply polynomials, following specific rules. The most common example is the ring of polynomials in one variable, denoted as R[x], where R represents the set of real numbers. Polynomial rings can also be defined over other sets, such as Z (integers) or F (finite fields). They are essential in various areas of mathematics, including algebra and number theory, as they help in solving equations and understanding algebraic structures.