PZP

pZ_p denotes the principal ideal generated by p in the ring Z_p of p-adic integers. The p-adic integers Z_p form the inverse limit of the finite rings Z/p^nZ and constitute a complete, compact topological ring that serves as the ring of integers in the p-adic number field Q_p.

The ideal pZ_p is the unique maximal ideal of Z_p, and Z_p is a discrete valuation ring with uniformizer p. The quotient Z_p / pZ_p is isomorphic to the finite field F_p. This makes Z_p a local ring with residue field F_p, and pZ_p the defining maximal ideal.

An element x in Z_p can be expressed by a convergent p-adic expansion x = a_0 + a_1 p + a_2 p^2 + ... with digits a_i in {0,1,...,p−1}. The element x lies in pZ_p precisely when the constant term a_0 equals 0, i.e., when v_p(x) ≥ 1, where v_p is the p-adic valuation. Units of Z_p are exactly the elements with v_p(x) = 0, equivalently those with a_0 ≠ 0.

Topology and structure: Z_p carries the p-adic topology, arising from the metric d(x,y) = p^{-v_p(x−y)}. Under this topology, Z_p is compact and totally disconnected; pZ_p is a closed subset and forms the maximal ideal.

Applications: pZ_p and Z_p play fundamental roles in local fields, ramification theory, modular reductions, and p-adic analytic methods. They are central to the study of Q_p and finite extensions, Hensel’s lemma, and arithmetic geometry.

See also: Z_p, Q_p, F_p, p-adic numbers.