2024/09/06 by Alizade, Rafail, Büyükaşık, Engin, Durgun, Yılmaz
#16D10 #16D90 #16P20 #16P40 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2409.04059
In this paper we study the modules M every simple subfactors of which is a homomorphic image of M and call them co-Kasch modules. These modules are dual to Kasch modules M every simple subfactors of which can be embedded in M. We show that a module is co-Kasch if and only if every simple module in σ[M] is a homomorphic image of M. In particular, a projective right module P is co-Kasch if and only if P is a generator for σ[P]. If R is right max and right H-ring, then every right R-module is co-Kasch; and the converse is true for the rings whose simple right modules have locally artinian injective hulls. For a right artinian ring R, we prove that: (1) every finitely generated right R-module is co-Kasch if and only if every right R-module is a co-Kasch module if and only if R is a right H-ring; and (2) every finitely generated projective right R-module is co-Kasch if and only if the Cartan matrix of R is a diagonal matrix. For a Prüfer domain R, we prove that, every nonzero ideal of R is co-Kasch if and only if R is Dedekind. The structure of ℤ-modules that are co-Kasch is completely characterized.