Y-Gorenstein Injective Module and Frobenius Bimodules

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

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Psq-injective modules and rings

Asian-European Journal of Mathematics ◽

10.1142/s1793557122501017 ◽

2021 ◽

pp. 2250101

Author(s):

Avanish Kumar Chaturvedi ◽

Sandeep Kumar

Keyword(s):

Injective Module ◽

Injective Modules

For any two right [Formula: see text]-modules [Formula: see text] and [Formula: see text], [Formula: see text] is said to be a ps-[Formula: see text]-injective module if, any monomorphism [Formula: see text] can be extended to [Formula: see text]. Also, [Formula: see text] is called psq-injective if [Formula: see text] is a ps-[Formula: see text]-injective module. We discuss some properties and characterizations in terms of psq-injective modules.

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New model structures and projective (injective) cotorsion pairs

Journal of Algebra and Its Applications ◽

10.1142/s0219498823500366 ◽

2021 ◽

Author(s):

Aimin Xu

Keyword(s):

Positive Integer ◽

Triangulated Categories ◽

Model Structure ◽

Cotorsion Pair ◽

Flat Dimension ◽

Gorenstein Projective Module ◽

Chain Complexes ◽

Cotorsion Pairs ◽

Gorenstein Injective ◽

Model Structures

Let [Formula: see text] be either the category of [Formula: see text]-modules or the category of chain complexes of [Formula: see text]-modules and [Formula: see text] a cofibrantly generated hereditary abelian model structure on [Formula: see text]. First, we get a new cofibrantly generated model structure on [Formula: see text] related to [Formula: see text] for any positive integer [Formula: see text], and hence, one can get new algebraic triangulated categories. Second, it is shown that any [Formula: see text]-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let [Formula: see text] be an [Formula: see text]-module with finite flat dimension and [Formula: see text] a positive integer, if [Formula: see text] is an exact sequence of [Formula: see text]-modules with every [Formula: see text] Gorenstein injective, then [Formula: see text] is injective.

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Example of an Injective Module which is Not Nice

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-028-4 ◽

1974 ◽

Vol 17 (1) ◽

pp. 133-134

Author(s):

Gerhard O. Michler

Keyword(s):

Artinian Ring ◽

Injective Module ◽

Finite Topology ◽

Factor Module ◽

Ring Of Quotients

In [1] Lambek calls the injective R-module I nice if every torsionfree factor module of the ring of quotients Q of R with respect to lis divisible. If lis nice then g is a dense subring of the bicommutator BicRI of I with respect to the finite topology (see [1, Proposition 2]). We now give an example of an injective R-module over an Artinian ring R which is not nice. Since R is Artinian, Q=BicRI, by Proposition B of [1].Before we give the example, we state the following, which depends on [2] for terminology.

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