A power-series ring is a mathematical term that refers to a commutative ring in which each element can be expressed as a power series. In simpler terms, it means that each element of the ring can be written as a sum of terms where each term contains a power of a fixed element. Some synonyms for a power-series ring include a formal power series ring, a power series extension, and a ring of formal power series. These terms convey the same meaning and are used interchangeably in mathematical literature. Examples of power-series rings include the ring of formal power series over a field and the ring of power series over a valuation ring. These rings have important applications in algebraic geometry and algebraic number theory.