Right orthogonal class of pure projective modules over pure hereditary rings

TL;DR

This paper studies the properties of $ ext{W}$-injective modules, a class of modules defined via pure projective modules, over pure hereditary rings, establishing existence, coresolution properties, and dimension bounds.

Contribution

It introduces and analyzes $ ext{W}$-injective modules over pure hereditary rings, proving existence of coresolutions, coresolving properties, and characterizing their dimensions.

Findings

Every module has a $ ext{W}$-injective preenvelope and coresolution.

The class of $ ext{W}$-injective modules is coresolving over pure-hereditary rings.

The $ ext{W}$-injective coresolution dimension is bounded and characterized by pure projective modules.

Abstract

We denote by $\mathcal{W}$ the class of all pure projective modules. Present article we investigate $\mathcal{W}$-injective modules and these modules are defined via the vanishing of cohomology of pure projective modules. First we prove that every module has a $\mathcal{W}$-injective preenvelope and then every module has a $\mathcal{W}$-injective coresolution over an arbitrary ring. Further, we show that the class of all $\mathcal{W}$-injective modules is coresolving (injectively resolving) over a pure-hereditary ring. Moreover, we analyze the dimension of $\mathcal{W}$-injective coresolution over a pure-hereditary ring. It is shown that $\sup\{ \cores_{\mathcal{W}^{\bot}}(M) \colon M \mbox{is an }R\mbox{-module }\} = \Fcor_{\mathcal{W}^{\bot}}(R) = \sup\{\pd(G) \colon G \mbox{ is a pure projective } R\mbox{-module}\}$ and we give some equivalent conditions of $\mathcal{W}$-injective envelope with the unique mapping property. In the last section, we proved the desirable properties of the dimension when the ring is semisimple artinian.