ZP2

Zp2 is a notation that can refer to more than one mathematical object, depending on context. The most common interpretations are the ring of integers modulo p^2 and, in other contexts, the direct product of two copies of the p-adic integers. Clarifying the intended meaning relies on the surrounding notation.

In modular arithmetic and number theory, Z/p^2Z (often written Z_{p^2} or Z/p^2Z) denotes the ring of integers modulo p^2. It consists of the p^2 residue classes {0, 1, ..., p^2−1} with addition and multiplication performed modulo p^2. The additive group is cyclic of order p^2, and the ring is local with a unique maximal ideal m = pZ/p^2Z generated by p. Elements coprime to p are precisely the units, and there are φ(p^2) = p^2 − p = p(p−1) of them. The element p is nilpotent (p^2 ≡ 0 mod p^2). Ideals form a chain 0 ⊂ (p) ⊂ Z/p^2Z, and the quotient Z/p^2Z / (p) is isomorphic to Z/pZ. This ring appears in lifting techniques such as Hensel’s lemma, modular computations, and constructions in algebraic number theory.

In other contexts, Zp2 may denote the direct product Z_p × Z_p, especially when Z_p denotes the p-adic integers. This object is a 2-dimensional module over Z_p and carries the product topology, making it a compact, totally disconnected topological ring. The notation Z_p^2 or Z_p ⊕ Z_p is common for this interpretation.

If you meant a specific Zp2, providing the context would help identify the intended meaning.