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Rachid Tribak - One of the best experts on this subject based on the ideXlab platform.
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Rad-⊕-Supplemented Modules and Cofinitely Rad-⊕-Supplemented Modules
Algebra Colloquium, 2012Co-Authors: Şule Ecevit, Muhammet T. Koşan, Rachid TribakAbstract:A module M is called (cofinitely) Rad-⊕-supplemented if every (cofinite) submodule of M has a Rad-supplement that is a Direct summand of M. We prove that if M is a coatomic cofinitely Rad-⊕-supplemented module, then M is an irredundant sum of local Direct Summands. We show that the classes of cofinitely Rad-⊕-supplemented modules and Rad-⊕-supplemented modules are closed under finite Direct sums. We also show that every Direct summand of a weak duo (cofinitely) Rad-⊕-supplemented module is (cofinitely) Rad-⊕-supplemented.
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Direct Summands of supplemented modules
Algebra Colloquium, 2007Co-Authors: Nil Orhan, Derya Keskin Tütüncü, Rachid TribakAbstract:A module M is called ⊕-supplemented if every submodule of M has a supplement that is a Direct summand of M. It is shown that if M is a ⊕-supplemented module and r(M) is a coclosed submodule of M for a left preradical r, then r(M) is a Direct summand of M, and both r(M) and M/r(M) are ⊕-supplemented.
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Direct Summands of ⊕-Supplemented Modules
Algebra Colloquium, 2007Co-Authors: Nil Orhan, Derya Keskin Tütüncü, Rachid TribakAbstract:A module M is called ⊕-supplemented if every submodule of M has a supplement that is a Direct summand of M. It is shown that if M is a ⊕-supplemented module and r(M) is a coclosed submodule of M for a left preradical r, then r(M) is a Direct summand of M, and both r(M) and M/r(M) are ⊕-supplemented.
Luis Núñez-betancourt - One of the best experts on this subject based on the ideXlab platform.
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Bernstein-Sato functional equations, $V$-filtrations, and multiplier ideals of Direct Summands
arXiv: Algebraic Geometry, 2019Co-Authors: Josep Àlvarez Montaner, Daniel J. Hernández, Jack Jeffries, Luis Núñez-betancourt, Pedro Teixeira, Emily E. WittAbstract:This paper investigates the existence and properties of a Bernstein-Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein-Sato polynomial for a Direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of $V$-filtrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible Summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular Direct Summands of polynomial rings.
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D-modules, Bernstein-Sato polynomials and F-invariants of Direct Summands
Advances in Mathematics, 2017Co-Authors: Josep Àlvarez Montaner, Craig Huneke, Luis Núñez-betancourtAbstract:Abstract We study the structure of D-modules over a ring R which is a Direct summand of a polynomial or a power series ring S with coefficients over a field. We relate properties of D-modules over R to D-modules over S. We show that the localization R f and the local cohomology module H I i ( R ) have finite length as D-modules over R. Furthermore, we show the existence of the Bernstein–Sato polynomial for elements in R. In positive characteristic, we use this relation between D-modules over R and S to show that the set of F-jumping numbers of an ideal I ⊆ R is contained in the set of F-jumping numbers of its extension in S. As a consequence, the F-jumping numbers of I in R form a discrete set of rational numbers. We also relate the Bernstein–Sato polynomial in R with the F-thresholds and the F-jumping numbers in R.
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$D$-modules, Bernstein-Sato polynomials and $F$-invariants of Direct Summands
arXiv: Commutative Algebra, 2016Co-Authors: Josep Àlvarez Montaner, Craig Huneke, Luis Núñez-betancourtAbstract:We study the structure of $D$-modules over a ring $R$ which is a Direct summand of a polynomial or a power series ring $S$ with coefficients over a field. We relate properties of $D$-modules over $R$ to $D$-modules over $S$. We show that the localization $R_f$ and the local cohomology module $H^i_I(R)$ have finite length as $D$-modules over $R$. Furthermore, we show the existence of the Bernstein-Sato polynomial for elements in $R$. In positive characteristic, we use this relation between $D$-modules over $R$ and $S$ to show that the set of $F$-jumping numbers of an ideal $I\subseteq R$ is contained in the set of $F$-jumping numbers of its extension in $S$. As a consequence, the $F$-jumping numbers of $I$ in $R$ form a discrete set of rational numbers. We also relate the Bernstein-Sato polynomial in $R$ with the $F$-thresholds and the $F$-jumping numbers in $R$.
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Local cohomology properties of Direct Summands
Journal of Pure and Applied Algebra, 2012Co-Authors: Luis Núñez-betancourtAbstract:Abstract In this article, we prove that if R → S is a homomorphism of Noetherian rings that splits, then for every i ≥ 0 and ideal I ⊂ R , Ass R H I i ( R ) is finite when Ass S H I S i ( S ) is finite. In addition, if S is a Cohen–Macaulay ring that is finitely generated as an R -module, such that all the Bass numbers of H I S i ( S ) , as an S -module, are finite, then all the Bass numbers of H I i ( R ) , as an R -module, are finite. Moreover, we show these results for a larger class a functors introduced by Lyubeznik [5] . As a consequence, we exhibit a Gorenstein F -regular UFD of positive characteristic that is not a Direct summand, not even a pure subring, of any regular ring.
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Local cohomology properties of Direct Summands
arXiv: Commutative Algebra, 2011Co-Authors: Luis Núñez-betancourtAbstract:In this article, we prove that if $R\to S$ is a homomorphism of Noetherian rings that splits, then for every $i\geq 0$ and ideal $I\subset R$, $\Ass_R H^i_I(R)$ is finite when $\Ass_S H^i_{IS}(S)$ is finite. In addition, if $S$ is a Cohen-Macaulay ring that is finitely generated as an $R$-module, such that all the Bass numbers of $H^i_{IS}(S)$, as an $S$-module, are finite, then all the Bass numbers of $H^i_{I}(R)$, as an $R$-module, are finite. Moreover, we show these results for a larger class a functors introduced by Lyubeznik. As a consequence, we exhibit a Gorenstein $F$-regular UFD of positive characteristic that is not a Direct summand, not even a pure subring, of any regular ring.
Molnár Lajos - One of the best experts on this subject based on the ideXlab platform.
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[TTK] Sequential isomorphisms between the sets of von Neumann algebra effects
2020Co-Authors: Molnár LajosAbstract:In this paper we describe the structure of all sequential isomorphisms between the sets of von Neumann algebra effects. It turns out that if the underlying algebras have no commutative Direct Summands, then every sequential isomorphism between the sets of their effects extends to the Direct sum of a *-isomorphism and a *-antiisomorphism between the underlying von Neumann algebras
Fatih Karabacak - One of the best experts on this subject based on the ideXlab platform.
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Generalized SSP-modules
Communications in Algebra, 2019Co-Authors: Özgür Taşdemir, Fatih KarabacakAbstract:We say an R-module M has the generalized summand sum property (GSSP), if the sum of any two Direct Summands is isomorphic to a Direct summand of M. This is dual notion to the generalized summand in...
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On SIP and SSP Modules
2018Co-Authors: Fatih KarabacakAbstract:M has the summmand intersection property (SIP), if the intersection of every two Direct Summands in M is a Direct summand in M, and a module has the summand sum property (SSP), if the sum of every two Direct summand in M is a Direct summand in M. In this note, we show that modules have these properties under some conditions.
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Generalized SIP-modules
Hacettepe Journal of Mathematics and Statistics, 2018Co-Authors: Özgür Taşdemir, Fatih KarabacakAbstract:We say an $R$-module $M$ has the generalized summand intersection property (briefly $GSIP$), if the intersection of any two Direct Summands is isomorphic to a Direct summand. This is a generalization of SIP modules. In this note, the characterization of this property over rings and modules is investigated and some useful propositions obtained in SIP modules are generalized to GSIP modules.
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On Generalizations of Extending Modules
Kyungpook mathematical journal, 2009Co-Authors: Fatih KarabacakAbstract:A module M is said to be SIP-extending if the intersection of every pair of Direct Summands is essential in a Direct summand of M. SIP-extending modules are a proper generalization of both SIP-modules and extending modules. Every Direct summand of an SIP-module is an SIP-module just as a Direct summand of an extending module is extending. While it is known that a Direct sum of SIP-extending modules is not neces- sarily SIP-extending, the question about Direct Summands of an SIP-extending module to be SIP-extending remains open. In this study, we show that a Direct summand of an SIP-extending module inherits this property under some conditions. Some related results are included about C11 and SIP-modules.
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ON MODULES AND MATRIX RINGS WITH SIP-EXTENDING
Taiwanese Journal of Mathematics, 2007Co-Authors: Fatih Karabacak, Adnan TercanAbstract:In this note we study modules with the property that the intersection of two Direct Summands is essential in a Direct summand (SIP-extending). Amongst other results we show that the class of right SIP-extending modules is neither closed under Direct sums nor Morita invariant. Further we deal with Direct Summands of a SIP-extending module and SIP-extending matrix rings.
Adnan Tercan - One of the best experts on this subject based on the ideXlab platform.
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When some complement of a z-closed submodule is a summand
Communications in Algebra, 2017Co-Authors: Yeliz Kara, Adnan TercanAbstract:ABSTRACTIn this article we study modules with the condition that every z-closed submodule has a complement which is a Direct summand. This new class of modules properly contains the class of extending modules. It is well known that the class of extending modules is closed under Direct Summands, but not under Direct sums. In contrast to extending (or CS) modules, it is shown that the class of modules with former property is closed under Direct sums. However we provide number of algebraic topological examples which show that this new class of modules is not closed under Direct Summands. To this end we obtain several results on the inheritance of the latter closure property.
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$PI$-extending modules via nontrivial complex bundles and Abelian endomorphism rings
Bulletin of The Iranian Mathematical Society, 2017Co-Authors: Yeliz Kara, Adnan Tercan, R. YaşarAbstract:A module is said to be $PI$-extending provided that every projection invariant submodule is essential in a Direct summand of the module. In this paper, we focus on Direct Summands and indecomposable decompositions of $PI$-extending modules. To this end, we provide several counter examples including the tangent bundles of complex spheres of dimensions bigger than or equal to 5 and certain hyper surfaces in projective spaces over complex numbers and obtain results when the $PI$-extending property is inherited by Direct Summands. Moreover, we show that under some module theoretical conditions $PI$-extending modules with Abelian endomorphism rings have indecomposable decompositions. Finally, we apply our former results, getting that, under suitable hypotheses, the finite exchange property implies the full exchange property.
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On the inheritance of the strongly π-extending property
Communications in Algebra, 2016Co-Authors: Yeliz Kara, Adnan TercanAbstract:ABSTRACTIn this article, we focus on modules with the property that every projection invariant submodule is essential in a fully invariant Direct summand. In contrast to π-extending condition, it is shown that the former property is inherited by Direct Summands and Morita invariant. An application of our results yields that the endomorphism ring of a free module enjoys the property. Moreover, we characterize generalized triangular matrix rings with the aforementioned property and apply to somewhat special cases.
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Modules Whose Submodules are Essentially Embedded in Direct Summands
Communications in Algebra, 2009Co-Authors: Fígen Takil, Adnan TercanAbstract:A module M is said to satisfy the C 12 condition if every submodule of M is essentially embedded in a Direct summand of M. It is known that the C 11 (and hence also C 1) condition implies the C 12 condition. We show that the class of C 12-modules is closed under Direct sums and also essential extensions whenever any module in the class is relative injective with respect to its essential extensions. We prove that if M is a -module with cancellable socle and satisfies ascending chain (respectively, descending chain) condition on essential submodules, then M is a Direct sum of a semisimple and a Noetherian (respectively, Artinian) submodules. Moreover, a C 12-module with cancellable socle is shown to be a Direct sum of a module with essential socle and a module with zero socle. An example is constructed to show that the reverse of the last result do not hold.
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ON MODULES AND MATRIX RINGS WITH SIP-EXTENDING
Taiwanese Journal of Mathematics, 2007Co-Authors: Fatih Karabacak, Adnan TercanAbstract:In this note we study modules with the property that the intersection of two Direct Summands is essential in a Direct summand (SIP-extending). Amongst other results we show that the class of right SIP-extending modules is neither closed under Direct sums nor Morita invariant. Further we deal with Direct Summands of a SIP-extending module and SIP-extending matrix rings.
