Moduloiden

Moduloiden is a term found in some algebraic literature to denote a class of structures that generalize modules by incorporating a modular congruence into their scalar action. In standard treatments, a moduloide over a ring R with an ideal I is a pair consisting of an abelian group M and an action of the quotient ring R/I on M, satisfying the usual module axioms: (r+I)·(m+n) = (r+I)·m + (r+I)·n, (r+I)·((s+I)·m) = (rs+I)·m, and 1+I acts as the identity on M. Equivalently, this is the same as giving M as an R-module on which I acts trivially (I·M = 0). Consequently, moduloiden are precisely modules over a quotient ring R/I.

Examples: If R = Z and I = nZ, then every Z/nZ-module M is a moduloide. A common case is M = Z/nZ itself, viewed as a module over Z/nZ by ordinary multiplication.

Substructures: A moduloide N ⊆ M is a subobject if N is a subgroup closed under the R/I-action. The quotient M/N inherits a natural R/I-module structure.

Morphisms: A moduloide homomorphism is a group homomorphism f: M → N that commutes with the R/I-action, i.e., f((r+I)·m) = (r+I)·f(m).

Relation to other notions: In practice, moduloiden reduce to ordinary modules over a quotient ring; the theory aligns with standard module theory over R/I.

Applications: Module-theoretic methods for finite abelian groups, coding theory, and algebraic number theory can be framed in terms of quotient-module structures.