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...

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

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...

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...

combines several modules into a new module. The most familiar examples of that construction occur in considering vector spaces, which are modules over a field...

(the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative...

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...

algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring...

particular, any non-trivial category with a zero object, such as an abelian category, is not Cartesian closed. So the category of modules over a ring is not Cartesian...

representation theory, the stable module category is a category in which projectives are "factored out." Let R be a ring. For two modules M and N over R, define...

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...

A} , there is an equivalence of categories from A {\displaystyle A} -modules to quasi-coherent sheaves, taking a module M {\displaystyle M} to the associated...

{\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...

R-Mod the category of left R-modules and by Mod-R the category of right R-modules (if R is commutative, the two categories coincide). Fix a module B in R-Mod...

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

this notion of product, Ab is a closed symmetric monoidal category. Ab is not a topos since e.g. it has a zero object. Category of modules Abelian sheaf...

a simple module is precisely a simple object in the category of (say left) modules. simplex category The simplex category Δ is the category where an object...

These notions of semi-simplicity can be unified using the language of semi-simple modules, and generalized to semi-simple categories. If one considers...

to the example of commutative rings above, one can show that all pullbacks exist in the category of groups and in the category of modules over some fixed...

regarded as Z-modules, so the category of abelian groups is also a symmetric, closed monoidal category. A symmetric compact closed category is a symmetric...

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked...

rings K-Vect, the category of vector spaces over a field K R-Mod, the category of modules over a commutative ring R CmptH, the category of all compact Hausdorff...

compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations...

role. The universality of this ring says that the category of representations of a Lie algebra is the same as the category of modules over its enveloping...

category C {\displaystyle {\mathcal {C}}} , and the category of modules over the ring is a category of functors defined on C {\displaystyle {\mathcal {C}}}...

R{\text{-Mod}}} be the category of modules over R {\displaystyle R} . (One can take this to mean either left R {\displaystyle R} -modules or right R {\displaystyle...

the category of topological spaces or categories of algebraic objects (like the category of groups, the category of rings, and the category of modules),...

abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categories of modules. This allows...