ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms...

then this is a module category in the old sense over the commutative ring OX(X). One can also consider modules over a semiring. Modules over rings are...

entire category of modules. Injective resolutions measure how far from injective a module is in terms of the injective dimension and represent modules in...

over R, is a R-algebra. the category of graded modules is a graded R-algebra. the category of chain complexes of R-modules is a differential graded algebra...

ring, then the category of finitely generated left modules over R is abelian. In particular, the category of finitely generated modules over a noetherian...

class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free...

sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the...

one can consider a category C enriched over the monoidal category of modules over a commutative ring R, called an R-linear category. In other words, each...

{\mathcal {A}}} be an abelian category. (Examples include the category of modules over a ring and the category of sheaves of abelian groups on a topological...

spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples...