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Keywords:
injective module; torsion submodule; local cohomology; $d$-transform
injective module; torsion submodule; local cohomology; $d$-transform
Summary:
Let $R$ be a commutative ring with identity and $M$ be an $R$-module. For a nonnegative integer $d$ and an ideal $\frak b$ of $R$, the concept of $(d,\frak b)$-injectivity in the category of $R$-modules is defined. We characterize the $(d,\frak b)$-injective hull of a module $M$ as a submodule of $E(M)$. Then we focus on the case when the ring $R$ is Noetherian and see the connection with some modules caused by some ideal transform functor $T_{(d,\frak b)}(-)$ and some local cohomology functors $H^i_{(d,\frak b)}(-)$ based on $(d,\frak b)$. As results we will see that over a Noetherian ring the functors $\Gamma _{(d,\frak b)}(-)$ and $T_{(d,\frak b)}(-)$ preserve the ${(d,\frak b)}$-injectivity. Among other results, for a $(d,\frak b)$-torsion free module $M$, we find a condition under which $T_{(d,\frak b)}(M)$ is injective.
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