Right orthogonal class of pure projective modules over pure hereditary rings

Abstract

Let $\mathcal{W}$ be the class of all pure projective modules. In this article, $\mathcal{W}$-injective modules is defined via the vanishing of cohomology of pure projective modules. First we show that every module has a $\mathcal{W}$-injective coresolution over an arbitrary ring and the class of all $\mathcal{W}$-injective modules is coresolving over a pure-hereditary ring. Further, we analyze the dimension of $\mathcal{W}$-injective coresolution over a pure-hereditary ring. It is shown that \Fcor_{\mathcal{W}^{\bot}}(R) = \sup\{\pd(G) \colon G is a pure projective $R$-module\} = \sup\{\cores_{\mathcal{W}^{\bot}}(M) \colon M is an $R$-module$\}.$ Finally, we give some equivalent conditions of $\mathcal{W}$-injective envelope with the unique mapping property. The dimension has desirable properties when the ring is semisimple artinian.Comment: 17 pages, 9 figure