Let K be a number field with ring of integers R. Let S be a finite set of prime ideals of R. An element x of K is an S-unit if the principal fractional ideal (x) is a product of primes in S (to positive or negative powers).
For the ring of rational integers Z one may take S to be a finite set of prime numbers and define an S-unit to be a rational number whose numerator and denominator are divisible only by the primes in S. The S-units form a multiplicative group containing the units of R. Dirichlet's unit theorem holds for S-units: the group of S-units is finitely generated, with rank (maximal number of multiplicatively independent elements) equal to r + s, where r is the rank of the unit group and s = |S|.
The S-unit equation is a Diophantine equation with u and v restricted to being S-units of K (or more generally, elements of a finitely generated subgroup of the multiplicative group of any field of characteristic zero).
The number of solutions of this equation is finite[1] and the solutions are effectively determined using estimates for linear forms in logarithms as developed in transcendental number theory.
A variety of Diophantine equations are reducible in principle to some form of the S-unit equation: a notable example is Siegel's theorem on integral points on elliptic curves, and more generally superelliptic curves of the form yn = f(x).