1. Modules: 4/2/13 1 2. Quotient Modules: 4/4/13 3 3. The Isomorphism Theorems: 4/9/13 5 4. Universal Properties: 4/11/13 7 5. The Tensor Product: 4/16/13 10 6. More Tensor Products: 4/18/13 12 7. Exact Sequences: 4/23/13 14 8. Projective and Injective Modules: 4/25/13 17 9. Finitely Generated Modules Over a Euclidean Domain: 4/30/13 19 10. Finitely Generated Modules Over a PID: 5/2/13 21 11. T...

‎In this paper we give an upper bound for Noetherian dimension of all injective modules with Krull dimension on arbitrary rings‎. ‎In particular‎, ‎we also give an upper bound for Noetherian dimension of all Artinian modules on Noetherian duo rings.

We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...

In 1966 [1], Auslander introduced a class of finitely generated modules having a certain complete resolution by projective modules. Then using these modules, he defined the G-dimension (G ostensibly for Gorenstein) of finitely generated modules. It seems appropriate then to call the modules of G-dimension 0 the Gorenstein projective modules. In [4], Gorenstein projective modules (whether finite...

Let $${\mathfrak {g}}$$ be a simple complex Lie algebra. In this paper we study the BGG category $${\mathcal {O}}_q$$ for quantum group $$U_q({\mathfrak {g}})$$ with q being root of unity in field K characteristic $$p >0$$ . We first consider modules and prove Steinberg tensor product theorem them. This result reduces problem determining corresponding irreducible characters to same finite subse...

The purpose of this paper is to describe a connection between model categories, a structure invented by algebraic topologists that allows one to introduce the ideas of homotopy theory to situations far removed from topological spaces, and cotorsion pairs, an algebraic notion that simultaneously generalizes the notion of projective and injective objects. In brief, a model category structure on a...