Right orthogonal class of pure projective modules over pure hereditary rings
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| Format: | Preprint |
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2016
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| _version_ | 1866910680802656256 |
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| author | Arunachalam, Umamaheswaran Ramalingam, Udhayakumar Chelliah, Selvaraj Venugopal, Shri Prakash |
| author_facet | Arunachalam, Umamaheswaran Ramalingam, Udhayakumar Chelliah, Selvaraj Venugopal, Shri Prakash |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1605_03704 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Right orthogonal class of pure projective modules over pure hereditary rings Arunachalam, Umamaheswaran Ramalingam, Udhayakumar Chelliah, Selvaraj Venugopal, Shri Prakash Rings and Algebras 18G05, 16D07, 16E10, 16E30, 18G10 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. |
| title | Right orthogonal class of pure projective modules over pure hereditary rings |
| topic | Rings and Algebras 18G05, 16D07, 16E10, 16E30, 18G10 |
| url | https://arxiv.org/abs/1605.03704 |