Self-Orthogonal Modules of Finite Projective Dimension |
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Authors: | Xiangyu Feng |
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Institution: | Department of Mathematics , Nanjing University , Nanjing, China |
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Abstract: | Let R be a ring and R ω a self-orthogonal module. We introduce the notion of the right orthogonal dimension (relative to R ω) of modules. We give a criterion for computing this relative right orthogonal dimension of modules. For a left coherent and semilocal ring R and a finitely presented self-orthogonal module R ω, we show that the projective dimension of R ω and the right orthogonal dimension (relative to R ω) of R/J are identical, where J is the Jacobson radical of R. As a consequence, we get that R ω has finite projective dimension if and only if every left (finitely presented) R-module has finite right orthogonal dimension (relative to R ω). If ω is a tilting module, we then prove that a left R-module has finite right orthogonal dimension (relative to R ω) if and only if it has a special ω⊥-preenvelope. |
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Keywords: | Projective dimension Right orthogonal dimension Self-orthogonal modules Tilting modules 𝒳 -coresolution dimension |
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