Rp×q

Rp×q is a notation used in mathematics to denote the Cartesian (or categorical) product of two algebraic structures called Rp and q. The exact meaning of Rp and q depends on context; they may be rings, groups, modules, or topological spaces. The product consists of all ordered pairs (r, s) with r in Rp and s in q, with operations defined componentwise: for rings, (r1, s1) + (r2, s2) = (r1 + r2, s1 + s2) and (r1, s1)(r2, s2) = (r1 r2, s1 s2). The same principle applies to other algebraic structures, and in topology Rp×q is given the product topology.

The product Rp×q satisfies the universal property of a categorical product: there are projection maps pi1: Rp×q → Rp and pi2: Rp×q → q, and for any object X with maps f: X → Rp and g: X → q there is a unique map h: X → Rp×q making the diagram commute (pi1 ∘ h = f and pi2 ∘ h = g).

In the ring case, ideals of Rp×q are precisely I × J where I is an ideal of Rp and J is an ideal of q. If Rp and q are finite rings, Rp×q is finite and its size is the product of the sizes. Note that Rp×q is not generally a field or an integral domain unless one factor is trivial. The notation is context dependent: Rp can denote a localization R_p, a p-adic-type object, or another ring; q is similarly flexible. In algebraic geometry or topology, Rp×q commonly appears as a simple product component, and it should not be confused with related constructions such as tensor products or direct sums.