WebJun 6, 2024 · $\begingroup$ Remark: another way to prove it is to use the facts that 1) every module is the direct limit of finitely presented modules and 2) direct limits are exact, if you already know these results. $\endgroup$ WebA commutative ring R has the property that every finitely presented module is a summand of a direct sum of cyclic modules if and only if Rm is a generalized valuation ring for each …

Is the kernel of a map to a finitely presented module finitely ...

WebThe R-module R/I is locally free since R is Boolean (and it is finitely generated as an R-module too, with a spanning set of size 1), but R/I is not projective because I is not a … WebIn algebra, a free presentation of a module M over a commutative ring R is an exact sequence of R-modules: Note the image under g of the standard basis generates M.In particular, if J is finite, then M is a finitely generated module.If I and J are finite sets, then the presentation is called a finite presentation; a module is called finitely presented if it … phenobarbital for dogs at hyvee

When is the category of finitely presented modules abelian?

WebDefinition. A module M over a ring R is flat if the following condition is satisfied: for every injective linear map: of R-modules, the map : is also injective, where is the map induced by ().. For this definition, it is enough to restrict the injections to the inclusions of finitely generated ideals into R.. Equivalently, an R-module M is flat if the tensor product with M … WebThe representation theory of such crossed products depends heavily on the image in the multiplicative group of F𝐹Fitalic_F of the 2-cocycle used to define the crossed product. WebLet. 0 \to M_1 \to M_2 \to M_3 \to 0. be a short exact sequence of R -modules. If M_1 and M_3 are finite R -modules, then M_2 is a finite R -module. If M_1 and M_3 are finitely … phenobarbital for asthma