Power Series Ring

The power series ring over a field is the ring of formal power series in the variables . Addition is coefficientwise, and multiplication uses the usual coefficient-convolution rule. Each product coefficient involves only finitely many summands, so no analytic convergence is required.

This is a complete local ring with maximal ideal and residue field . It is a regular local ring of Krull dimension (Bruns and Herzog 1998). The quotient by retains exactly the monomials of total degree below , which gives Hilbert-Samuel multiplicity 1.