Yazar "Ertas, Nil Orhan" seçeneğine göre listele

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    Direct sums of ADS* modules
    (Springer Basel Ag, 2016) Tribak, Rachid; Tutuncu, Derya Keskin; Ertas, Nil Orhan
    A module M is called ADS* if for every direct summand N of M and every supplement K of N in M, we have M = N circle plus K. In this work, we study direct sums of ADS* modules. Many examples are provided to show that this notion is not inherited by direct sums. It is shown that if a module M has a decomposition M = A circle plus B which complements direct summands such that A and B are mutually projective, then M is ADS*. The class of rings R, for which all direct sums of ADS* R-modules are ADS*, is shown to be exactly that of the right V-rings. We characterize the class of right perfect rings R for which R circle plus S is ADS* for every simple R-module S as that of the semisimple rings.
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    MIXED INJECTIVE MODULES
    (Cambridge Univ Press, 2010) Tutuncu, Derya Keskin; Mohamed, Saad H.; Ertas, Nil Orhan
    Since Azumaya introduced the notion of A-injectivity in 1974, several generalizations have been investigated by a number of authors. We introduce some more generalizations and discuss their connection to the previous ones.
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    On dual Rickart modules and weak dual Rickart modules
    (Luhansk Taras Shevchenko Natl Univ, 2018) Tutuncu, Derya Keskin; Ertas, Nil Orhan; Tribak, Rachid
    Let R be a ring. A right R-module M is called d-Rickart if for every endomorphism phi of M, phi(M) is a direct summand of M and it is called wd-Rickart if for every nonzero endomorphism phi of M, phi(M) contains a nonzero direct summand of M. We begin with some basic properties of (w)d-Rickart modules. Then we study direct sums of (w)d-Rickart modules and the class of rings for which every finitely generated module is (w)d-Rickart. We conclude by some structure results.
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    ON FULLY IDEMPOTENT MODULES
    (Taylor & Francis Inc, 2011) Tutuncu, Derya Keskin; Ertas, Nil Orhan; Tribak, Rachid; Smith, Patrick F.
    A submodule N of a module M is idempotent if N = Hom (M,N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.
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    On weak Rickart modules
    (World Scientific Publ Co Pte Ltd, 2017) Tutuncu, Derya Keskin; Ertas, Nil Orhan; Tribak, Rachid
    We introduce and study the notion of w-Rickart modules (i.e. modules M such that for every nonzero endomorphism phi of M, the kernel of phi is contained in a proper direct summand of M). We show that the class of right w-Rickart modules lies properly between the class of right K-nonsingular modules and the class of right Rickart modules. Many properties of w-Rickart modules are established. Some relevant counterexamples are indicated.
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    RINGS WHOSE CYCLIC MODULES ARE DIRECT SUMS OF EXTENDING MODULES
    (Cambridge Univ Press, 2012) Aydogdu, Pinar; Er, Noyan; Ertas, Nil Orhan
    Dedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: Acyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (not congruent to R-R) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and boolean AND(n is an element of N)J(n) = J(m) for some m is an element of N if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.
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    Rings whose modules have maximal or minimal projectivity domain
    (Elsevier, 2012) Holston, Chris; Lopez-Permouth, Sergio R.; Ertas, Nil Orhan
    Using the notion of relative projectivity, projective modules may be thought of as being those which are projective relative to all others. In contrast, a module M is said to be projectively poor if it is projective relative only to semisimple modules. We prove that all rings have projectively poor modules. In fact, every ring even has a semisimple projectively poor module. Properties of projectively poor modules are studied and particular emphasis is given to the study of modules over PCI domains; we note that over such domains when all right ideals are principal most modules seem to be either projective or projectively poor. We consider rings over which modules are either projective or projectively poor and call them rings without a p-middle class. We show that a QF ring R with homogeneous right socle and J(R)(2) = 0 has no right p-middle class. As we analyze the structure of rings with no right p-middle class, among other results, we show that any such ring is the ring direct sum of a semisimple artinian ring and a ring K which is either zero or an indecomposable ring such that either (i) K is a semiprimary right SI-ring with J(K) not equal 0, or (ii) K is a semiprimary ring with Soc(K-K) = Z(r)(K) = J(K) not equal 0, or (iii) K is a prime ring with Soc(K-K) = 0, and either f (K) = 0 or (K)J(K) and J(K)(K) are infinitely generated, or (iv) K is a prime right SI-ring with infinitely generated right socle. (C) 2011 Elsevier B.V. All rights reserved.
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    Some properties of intersection graph of a module with an application of the graph of DOUBLE-STRUCK CAPITAL Zn
    (Taylor & Francis Ltd, 2021) Ertas, Nil Orhan; Surul, Sema
    Let R be an unital ring which is not necessarily commutative. The intersection graph of ideals of R is a graph with the vertex set which contains proper ideals of R and distinct two vertices I and J are adjacent if and only if I c J (1) 0 is denoted by G. In this paper, we will give some properties of regular graph, triangle-free graph and clique number of G(M) for a module M. We also characterize girth of an Artinian module with connected module. We characterize the chromatic number of G(Z (n) ). We also give an algorithm for the chromatic number of G(Z (n) ).
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    A Variation of Coretractable Modules
    (Malaysian Mathematical Sciences Soc, 2018) Ertas, Nil Orhan; Tutuncu, Derya Keskin; Tribak, Rachid
    A right R-module M is called coretractable (s-coretractable) if Hom(M/K, M) not equal 0 for any proper submodule (supplement submodule) K of M. In this article, we continue the study of coretractable modules. Then we study s-coretractable modules. It is shown that this property is not inherited by direct summands and a direct sum of s-coretractable modules may not be s-coretractable. Examples are provided to illustrate and delineate the results.