moduly

Moduly, in mathematical usage, are algebraic structures that generalize vector spaces by allowing scalars from a ring rather than from a field. An R-module M consists of an abelian group (M, +) together with an action R × M → M that satisfies distributivity and associativity: (r + s)m = rm + sm, r(m + n) = rm + rn, (rs)m = r(sm), and 1R m = m when the ring R has a multiplicative identity. Subsets of M closed under addition and scalar multiplication are called submodules. A homomorphism between R-modules preserves both the additive structure and the scalar action, and kernels and images play the usual role in studying these maps. Quotient modules M/N are formed from submodules N.

Examples help situate the concept. A module over the integers Z is the same thing as an abelian group. A vector space over a field F is an F-module; modules over a field satisfy stronger division properties. The free module R^n consists of n-tuples of elements of R with componentwise addition and scalar multiplication, and serves as a natural example when a basis exists. Not every module has a basis, but a module with a basis is called free and is isomorphic to a direct sum of copies of R.

Key constructions include direct sums, tensor products, and Hom-spaces, which organize modules into a rich category with homological tools. Finitely generated modules and their structure are central in algebra: over a principal ideal domain, the structure theorem classifies finitely generated modules up to isomorphism as a direct sum of cyclic modules.

Moduly form a foundational framework across algebra, representation theory, algebraic topology and algebraic geometry, providing a language to study linear-like structures over rings.