Let T be a positive integer and M be a finitely presented module over the Laurent polynomial ring ℤ/T[X1±, …, XN±]. We consider S-unit equations over M: these are equations of the form x1 m1 + ⋯ + xK mK = m0, where the variables x1, …, xK range over the set of monomials (with coefficient 1) of ℤ/T[X1±, …, XN±]. When T is a power of a prime number p, we show that the solution set of an S-unit equation over M is effectively p-normal in the sense of Derksen and Masser (2015). This generalizes their result on S-unit equations in fields of prime characteristic. When T is an arbitrary positive integer, we show that deciding whether an S-unit equation over M admits a solution is Turing equivalent to solving a system of linear-exponential Diophantine equations, whose base contains the prime divisors of T. Combined with a recent result of Karimov, Luca, Nieuwveld, Ouaknine and Worrell (2025), this yields decidability when T has at most two distinct prime divisors. This also shows that proving either decidability or undecidability in the case of arbitrary T would entail major breakthroughs in number theory. S-unit equations in modules have direct connections to many problems in computational algebra such as finding sparse polynomials in ideals, identifying zeros of linear recurrence sequences, and deciding membership problems in metabelian groups. In particular, a direct consequence of our result is the decidability Submonoid Membership in wreath products of the form ℤ/pa qb ≀ ℤd.

We show that the Skolem Problem is decidable in finitely generated commutative rings of positive characteristic. More precisely, we show that there exists an algorithm which, given a finite presentation of a (unitary) commutative ring R = ℤ/T[X1, …, Xn]/I of characteristic T > 0, and a linear recurrence sequence (γn)n ∈ ℕ ∈ Rℕ, determines whether (γn)n ∈ ℕ contains a zero term. Our proof is based on two recent results: Dong and Shafrir (2026) on the solution set of S-unit equations over pe-torsion modules, and Karimov, Luca, Nieuwveld, Ouaknine, and Worrell (2025) on solving linear equations over powers of two multiplicatively independent numbers. Our result implies, moreover, that the zero set of a linear recurrence sequence over a ring of characteristic T = p1e1 ⋯ pkek is effectively a finite union of pi-normal sets in the sense of Derksen (2007).