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Let lamda be an associative ring with identity and with the Jacobson radical r, let mod lamda be the category of finitely generated left lamda-modules, and let lamdaop be the opposite ring of lamda. All modules are left unital modules, and if X is a module then pd X is the projective dimension of X. If lamda is left artinian and M in mod labmda, we denote by P(M) a projective cover of M.

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