Tensor Product of $C$-Injective Modules

TL;DR

This paper investigates the conditions under which the tensor product of $C$-injective modules remains $C$-injective and characterizes when the torsion product is $C$-injective, extending known theorems in module theory.

Contribution

It establishes new criteria linking $C$-injectivity of tensor and torsion products to properties of the semidualizing module $C$, generalizing previous results.

Findings

Tensor product of $C$-injective modules is $C$-injective iff the injective hull of $C$ is $C$-flat.

$C$ is pointwise dualizing iff $Tor^R_i(M,N)$ is $C$-injective for all $C$-injective modules.

Results recover and extend theorems of Enochs and Jenda.

Abstract

Let $R$ be a Noetherian ring and let $C$ be a semidualizing $R$-module. In this paper, we are concerned with the tensor and torsion product of $C$-injective modules. Firstly, it is shown that the tensor product of any two $C$-injective $R$-modules is $C$-injective if and only if the injective hull of $C$ is $C$-flat. Secondly, it is proved that $C$ is a pointwise dualizing $R$-module if and only if $Tor^R_i(M,N)$ is $C$-injective for all $C$-injective $R$-modules $M$ and $N$, and all $ i \geq 0$. These results recover the celebrated theorems of Enochs and Jenda \cite{EJ2}.