Let $R$ be semiperfect commutative Noetherian ring and $C$ be a semidualizing $R$--module. The co... more Let $R$ be semiperfect commutative Noetherian ring and $C$ be a semidualizing $R$--module. The connection of the Serre condition $(S_n)$ on a horizontally linked $R$-module of finite $\gc$-dimension with the vanishing of certain cohomology modules of its linked module is discussed. As a consequence, it is shown that under some conditions Cohen-Macaulayness is preserved under horizontally linkage.
Inspired by Buchweitz and Flenner [3], we show that, for a semidualizing bimodule C, C-perfect co... more Inspired by Buchweitz and Flenner [3], we show that, for a semidualizing bimodule C, C-perfect complexes have the ability to detect when a ring is strongly regular. It is shown that there exists a class of modules which admit minimal resolutions of C-projective modules.
Inspired by Jorgensen et. al., it is proved that if a Cohen--Macaulay local ring $R$ with dualizi... more Inspired by Jorgensen et. al., it is proved that if a Cohen--Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$--modules of length $n$, then $R\cong Q/(I_1+\cdots+I_n)$ for some Gorenstein ring $Q$ and ideals $I_1,\cdots, I_n$ of $Q$; and, for each $\Lambda\subseteq [n]$, the ring $Q/(\Sigma_{l\in \Lambda} I_l)$ has some interesting cohomological properties . This extends the result of Jorgensen et. al., and also of Foxby and Reiten.
The concept of a sequence of exact zero-divisors on a noetherian local ring is defined and studie... more The concept of a sequence of exact zero-divisors on a noetherian local ring is defined and studied. Some properties of sequences of exact zero-divisors are compared with regular sequences.
Let ${\left( {R,\mathfrak{m}} \right)}$ be a commutative Noetherian local ring and let $\mathfrak... more Let ${\left( {R,\mathfrak{m}} \right)}$ be a commutative Noetherian local ring and let $\mathfrak{a}$ be an ideal of R. We give some inequalities between the Bass numbers of an R–module and those of its local cohomology modules with respect to $\mathfrak{a}$ . As an application of these inequalities, we recover results of Delfino-Marley and Kawasaki by showing that for a minimax R-module M and for any non-negative integer i, the Bass numbers of the ith local cohomology module ${\text{H}}^{i}_{\mathfrak{a}} {\left( M \right)}$ are finite if one of the following holds: $R \mathord{\left/ {\vphantom {R \mathfrak{a}}} \right. \kern-0em} \mathfrak{a} = 1$ , $\mathfrak{a}$ is a principal ideal.
Let $R$ be semiperfect commutative Noetherian ring and $C$ be a semidualizing $R$--module. The co... more Let $R$ be semiperfect commutative Noetherian ring and $C$ be a semidualizing $R$--module. The connection of the Serre condition $(S_n)$ on a horizontally linked $R$-module of finite $\gc$-dimension with the vanishing of certain cohomology modules of its linked module is discussed. As a consequence, it is shown that under some conditions Cohen-Macaulayness is preserved under horizontally linkage.
Inspired by Buchweitz and Flenner [3], we show that, for a semidualizing bimodule C, C-perfect co... more Inspired by Buchweitz and Flenner [3], we show that, for a semidualizing bimodule C, C-perfect complexes have the ability to detect when a ring is strongly regular. It is shown that there exists a class of modules which admit minimal resolutions of C-projective modules.
Inspired by Jorgensen et. al., it is proved that if a Cohen--Macaulay local ring $R$ with dualizi... more Inspired by Jorgensen et. al., it is proved that if a Cohen--Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$--modules of length $n$, then $R\cong Q/(I_1+\cdots+I_n)$ for some Gorenstein ring $Q$ and ideals $I_1,\cdots, I_n$ of $Q$; and, for each $\Lambda\subseteq [n]$, the ring $Q/(\Sigma_{l\in \Lambda} I_l)$ has some interesting cohomological properties . This extends the result of Jorgensen et. al., and also of Foxby and Reiten.
The concept of a sequence of exact zero-divisors on a noetherian local ring is defined and studie... more The concept of a sequence of exact zero-divisors on a noetherian local ring is defined and studied. Some properties of sequences of exact zero-divisors are compared with regular sequences.
Let ${\left( {R,\mathfrak{m}} \right)}$ be a commutative Noetherian local ring and let $\mathfrak... more Let ${\left( {R,\mathfrak{m}} \right)}$ be a commutative Noetherian local ring and let $\mathfrak{a}$ be an ideal of R. We give some inequalities between the Bass numbers of an R–module and those of its local cohomology modules with respect to $\mathfrak{a}$ . As an application of these inequalities, we recover results of Delfino-Marley and Kawasaki by showing that for a minimax R-module M and for any non-negative integer i, the Bass numbers of the ith local cohomology module ${\text{H}}^{i}_{\mathfrak{a}} {\left( M \right)}$ are finite if one of the following holds: $R \mathord{\left/ {\vphantom {R \mathfrak{a}}} \right. \kern-0em} \mathfrak{a} = 1$ , $\mathfrak{a}$ is a principal ideal.
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