Nonzero Direct Summand Research Articles

On strongly coseparable modules

A module M is called mathfrak {s}-coseparable if for every nonzero submodule U of M such that M/U is finitely generated, there exists a nonzero direct summand V of M such that V subseteq U and M/V is finitely generated. It is shown that every non-finitely generated free module is mathfrak {s}-coseparable but a finitely generated free module is not, in general, mathfrak {s}-coseparable. We prove that the class of mathfrak {s}-coseparable modules over a right noetherian ring is closed under finite direct sums. We show that the class of commutative rings R for which every cyclic R-module is mathfrak {s}-coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is mathfrak {s}-coseparable are provided.

Modules of infinite regularity over commutative graded rings

In this work, we prove that if a graded, commutative algebra $R$ over a field $k$ is not Koszul then, denoting by $\mathfrak{m}$ the maximal homogeneous ideal, the nonzero modules of the form $\mathfrak{m} M$ have infinite Castelnuovo-Mumford regularity. We also prove that over complete intersections which are not Koszul, a nonzero direct summand of a syzygy of $k$ has infinite regularity. Finally, we relate the vanishing of the graded deviations of $R$ to having a nonzero direct summand of a syzygy of $k$ of finite regularity.

Characterizations of regular local rings via syzygy modules of the residue field

Let $R$ be a commutative Noetherian local ring with residue field $k$. We show that if a finite direct sum of syzygy modules of $k$ surjects onto `a semidualizing module' or `a non-zero maximal Cohen-Macaulay module of finite injective dimension', then $R$ is regular. We also prove that $R$ is regular if and only if some syzygy module of $k$ has a non-zero direct summand of finite injective dimension.

A note on generalizations of quasi-Frobenius rings

A ring [Formula: see text] is called quasi-Frobenius, briefly QF, if [Formula: see text] is right (or left) Artinian and right (or left) self-injective. A ring [Formula: see text] is called right co-Harada if every noncosmall right [Formula: see text]-module contains a nonzero projective direct summand and [Formula: see text] satisfies the ACC on right annihilators. The class of co-Harada rings is one of the most interesting generalizations of QF rings. When considering relation between these ring classes, [K. Oshiro, Lifting modules, extending modules and their applications to QF-ring, Hokkaido Math. J. 13 (1984) 310–338, Theorem 4.3] showed that a ring [Formula: see text] is QF if and only if it is right co-Harada ring with [Formula: see text]. In this note, we show that a ring [Formula: see text] is QF if and only if it is right co-Harada ring and satisfies either [Formula: see text] or [Formula: see text].

RETRACTION NOTE TO: X-LIFTING MODULES OVER RIGHT PERFECT RINGS

Keskin and Harmanci defined the family B(M, X) = {A ≤ M | ∃Y ≤ X, ∃f ∈ HomR(M, X/Y ), Ker f/A ? M/A}. And Orhan and Keskin generalized projective modules via the class B(M, X). In this note we introduce X-local summands and X-hollow modules via the class B(M, X). Let R be a right perfect ring and let M be an X-lifting module. We prove that if every co-closed submodule of any projective module P contains Rad(P ), then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang’s result [9, Theorem 3.4]. Let X be an R-module. We also prove that for an X-hollow module H such that every non-zero direct summand K of H with K ∈ B(H, X), if H ⊕H has the internal exchange property, then H has a local endomorphism ring.

On Weak Dual Rickart Modules and Dual Baer Modules

We introduce and study the notion of wd-Rickart modules (i.e. modules M such that for every nonzero endomorphism ϕ of M, the image of ϕ contains a nonzero direct summand of M). We show that the class of rings R for which every right R-module is wd-Rickart is exactly that of right semi-artinian right V-rings. We prove that a module M is dual Baer if and only if M is wd-Rickart and M has the strong summand sum property. Several structure results for some classes of wd-Rickart modules and dual Baer modules are provided. Some relevant counterexamples are indicated.

X-LIFTING MODULES OVER RIGHT PERFECT RINGS

Keskin and Harmanci defined the family B(M, X) = {A ≤ M | ∃Y ≤ X, ∃f ∈ HomR(M, X/Y ), Ker f/A ? M/A}. And Orhan and Keskin generalized projective modules via the class B(M, X). In this note we introduce X-local summands and X-hollow modules via the class B(M, X). Let R be a right perfect ring and let M be an X-lifting module. We prove that if every co-closed submodule of any projective module P contains Rad(P ), then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang’s result [9, Theorem 3.4]. Let X be an R-module. We also prove that for an X-hollow module H such that every non-zero direct summand K of H with K ∈ B(H, X), if H ⊕H has the internal exchange property, then H has a local endomorphism ring.

The Center Of A Generalized Effect Algebra

AbstractIn this article, we study the center of a generalized effect algebra (GEA), relate it to the exocenter, and in case the GEA is centrally orthocomplete (a COGEA), relate it to the exocentral cover system. Our main results are that the center of a COGEA is a complete boolean algebra and that a COGEA decomposes uniquely as the direct sum of an effect algebra (EA) that contains the center of the COGEA and a complementary direct summand in which no nonzero direct summand is an EA.

A CHARACTERIZATION OF CO-HARADA RING

A ring R is called right co-Harada if every non-cosmall right R-module contains a non-zero projective direct summand and R satisfies the ACC on right annihilators. In this paper we give a characterization of co-Harada rings via left perfect rings.

Integral Lie powers of a lattice for the cyclic group of order 2

Let L n denote the n-th homogeneous component of the free Lie ring L(W) on a given \({{\Bbb Z}}C_{2}\)-lattice W. This paper gives explicit formulae for the multiplicities of the three indecomposable \({{\Bbb Z}}C_{2}\)-lattices in a Krull-Schmidt decomposition of L n . In the case where W is a free \({{\Bbb Z}}C_{2}\)-lattice, L n is shown to have no non-zero direct summand on which C 2 acts trivially – this extends a result of R. M. Bryant for the special case where W is the regular \({{\Bbb Z}}C_{2}\)-lattice. As an application, the structure of the higher dimensional modules associated to a non-cyclic free presentation of C 2 is determined.

Bouquets of Baer modules

In this paper some transcendental numbers are used to construct infinite-dimensional indecomposable Baer modules. Let R be a ring whose category of modules has a torsion theory. An R-module, M, is Baer if every extension of M by any torsion R-module splits. In this paper, R will be a path algebra, i.e., an algebra whose basis over a field K are the vertices and paths of a directed graph. Multiplication is given by path composition. When R is a path algebra obtained from an extended Coxeter-Dynkin diagram with no oriented cycles, we characterize Baer modules of countable rank. This characterization is used to show that modules constructed from Liouville sequences yield a family, ▪ = { B n } ∞ n=0 , of Baer modules satisfying the following conditions: every extension of B m by B n splits for every pair ( m,n); if m≠ n, B m is not isomorphic to B n , while automorphisms of B n are given by multiplications by nonzero elements of K. Each B n is shown to be a submodule of a rank-one module. Another application of our characterization is the determination of the rank-one modules with the property that every submodule of infinite rank has a nonzero direct summand that is Baer. In analogy with ℵ r -free modules, we define ℵ r -Baer modules and give an example of an ℵ 1-Baer module that is not Baer. The existence of a Baer module, M, that is not a direct sum of Baer modules of countable rank is also proved. However every nonzero submodule of M has a nonzero direct summand. A problem suggested by these results is the existence and structure of indecomposable Baer modules of uncountable rank.

Tidy Noether lattices

A Noether lattice ℒ satisfying the union condition on primes which is not a domain and in which every nonzero principal element is integrally closed is characterized in terms of its direct summands. It is shown that either: (1) ifℒ has no proper nonzero direct summands, then every nonzero principal element ofℒ is integrally closed if and only if ℒ is a local Noether lattice whose maximal element is principal and has square zero; or (2) ifℒ has a proper nonzero direct summand, then every nonzero principal element ofℒ is integrally closed if and only if for each minimal direct summandA ofℒ, the quotient lattice [0,A] is an integrally closed domain.

THE LARGEST NON-TRIVIAL TORSION CLASS OF MODULES

ABSTRACT In this paper we investigate the following two classes of left R-modules: N(P) ={A|A has no non-zero direct summand P e P} and H(p) = {A} if B ⋚ A with B e N(P), then B = 0}, where P is a class of projective R-modules. We demonstrate that N(p) is, in general, not a torsion class but that H(P) is always a torsionfree class. We also investigate those classes P and rings R for which N(P) is the largest non-trivial torsion class of R-modules.