Module

An algebraic structure consisting of an abelian group (M, +) together with an action of a ring R (not necessarily commutative or a field) on M that is distributive and associative with respect to ring multiplication and respects the ring identity if present; modules generalize vector spaces by allowing scalars from rings.

A structure M equipped with an abelian group operation (addition) and an action of a ring R (scalar multiplication) that is compatible with ring multiplication and addition: r·(m+n) = r·m + r·n, (r+s)·m = r·m + s·m, and (rs)·m = r·(s·m).

An R-module is an abelian group equipped with a compatible action of a ring R: a map R × M → M satisfying distributivity, associativity with ring multiplication, and that 1·m = m when R has unity. It generalizes vector spaces by allowing scalars from a ring.

An algebraic structure consisting of an abelian group (the underlying additive group) together with an action of a ring R (with unity) on that group satisfying distributivity, associativity with ring multiplication, and identity action; a module is a direct generalization of a vector space where scalars come from a ring instead of a field.