Formal power series ring

Definition

Let $R$ be a commutative unital ring. The formal power series ring over $R$ in one variable, denoted $R[[x]]$ if the variable (indeterminate) is $x$ is the ring whose elements are possibly infinite formal linear combinations of nonnegative integral powers of $x$, with addition coordinate-wise and multiplication extended $R$-linearly (infinitely so) from a multiplication of powers that adds up the exponents.

Related notions

• Laurent series ring
• Puiseux series ring
• Hahn series ring