ZppZp

ZppZp is a theoretical construct used in algebraic number theory to illustrate how p-adic integers relate to modular reduction through a fiber product construction. It is not a standardly named object with a universal definition, but a descriptive label used in expository contexts to explore lifting properties and compatibility between p-adic and modulo p structures.

Definition and basic structure

Let p be a prime and Z_p denote the ring of p-adic integers. Let ρ: Z_p → Z_p/pZ_p ≅ Z/pZ be reduction modulo p. The ZppZp construction is defined as the fiber product Z_p ×_{Z_p/pZ_p} Z_p, consisting of pairs (a,b) in Z_p × Z_p such that ρ(a) = ρ(b); equivalently, a ≡ b (mod p). The ring ZppZp comes with natural projections π1: ZppZp → Z_p and π2: ZppZp → Z_p, and a diagonal embedding Δ: Z_p → ZppZp given by a ↦ (a,a).

Properties and interpretation

ZppZp is a commutative ring with unity, and it sits as a subring of Z_p × Z_p. Its elements encode two p-adic integers that lift the same residue class modulo p. The ring generally contains zero divisors; for example, (p,0) is a nonzero element of ZppZp, while (p,0)·(0,1) = (0,0). The construction provides a concrete model for studying how congruence modulo p interacts with the richer structure of p-adic integers.

Context and usage

In educational texts and problem sets, ZppZp serves as a simple example of a fiber product in a p-adic setting. The notation is not universally standard, and the same idea may be described with different terminology depending on the author.

See also

Z_p, p-adic integers, Z/pZ, fiber product, ring theory.