Smodules

Smodules are algebraic structures that generalize modules by incorporating a secondary symmetry given by a monoid action. An S-module consists of a left module M over a ring R together with an action of a monoid S on M that respects the R-module structure.

Formally, let R be a ring, M an abelian group with a left R-action making it an R-module, and S a monoid. A left S-action on M is a monoid homomorphism ρ: S → End_R(M), where End_R(M) denotes the set of R-linear endomorphisms of M. Then (M, ρ) is called an S-module over R. The action is compatible in the sense that each φ_s = ρ(s) is R-linear, and φ_s(r m) = r φ_s(m) for all r ∈ R, m ∈ M, s ∈ S.

Morphisms between S-modules are R-linear maps that commute with the S-action: a map f: M → N is an S-module homomorphism if f(r m) = r f(m) for all r, m and f ∘ φ_s^M = φ_s^N ∘ f for all s ∈ S.

Relation to other constructions: if S acts on M via endomorphisms, M can be viewed as a module over the monoid ring R[S], making S-modules a representation theory object for the monoid S. If S is trivial, one recovers ordinary R-modules. If S is a group, S-modules specialize to representations of the group on R-modules. There are variations where S also acts on R, leading to skew or crossed product constructions.

Overview and use: Smodules provide a framework to study modules with internal symmetries or time evolutions, appearing in representation theory, invariant theory, and algebraic dynamics. The terminology S-module is used variably in literature, sometimes overlapping with modules over monoid rings.