Browsing by Author "Kurtulmaz, Yosum"

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    Almost unit-clean rings
    (Editura Academiei Romane, 2019) Chen, H.; Köse, H.; Kurtulmaz, Yosum
    A ring R is almost unit-clean provided that every element in R is equivalent to the sum of an idempotent and a regular element. We investigate conditions under which a ring is almost unit-clean. We prove that every ring in which every zero-divisor is strongly _-regular is almost unit-clean and every matrix ring of elementary divisor domains is almost unit-clean. Furthermore, it is shown that the trivial extension R(M) of a commutative ring R and an R-module M is almost unit-clean if and only if each x 2 R can be written in the form ux = r+e where u 2 U(R); r 2 R 􀀀 (Z(R) [ Z(M)) and e 2 Id(R). We thereby construct many examples of such rings.
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    Certain clean decompositions for matrices over local rings
    (Kyungpook National University Department of Mathematics, 2023-12-31) Kurtulmaz, Yosum ; Köse, Handan; Chen, Huanyin
    An element a E R is strongly rad-clean provided that there exists an idempotent e E R such that a -e E U(R), ae = ea and eae E J(eRe). In this article, we completely determine when a 2 x 2 matrix over a commutative local ring is strongly rad clean. An application to matrices over power-series is also given.
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    Duo property for rings by the quasinilpotent perspective
    (Vasyl Stefanyk Precarpathian National University, 2021-03-16) Harmancı, A.; Kurtulmaz, Yosum; Üngör, B.
    In this paper, we focus on the duo ring property via quasinilpotent elements, which gives anew kind of generalizations of commutativity. We call this kind of ringsqnil-duo. Firstly, someproperties of quasinilpotents in a ring are provided. Then the set of quasinilpotents is applied tothe duo property of rings, in this perspective, we introduceand study right (resp., left) qnil-duorings. We show that this concept is not left-right symmetric. Among others, it is proved that if theHurwitz series ringH(R;α)is right qnil-duo, thenRis right qnil-duo. Every right qnil-duo ring isabelian. A right qnil-duo exchange ring has stable range 1.
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    Local comparability of exchange ideals
    (Hacettepe University, 2019) Köse, H.; Kurtulmaz, Yosum; Chen, H.
    An exchange ideal I of a ring R is locally comparable if for every regular x ∈ I there exists a right or left invertible u ∈ 1+I such that x = xux. We prove that every matrix extension of an exchange locally comparable ideal is locally comparable. We thereby prove that every square regular matrix over such ideal admits a diagonal reduction.
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    A nil approach to symmetricity of rings
    (Allahabad Mathematical Society, 2018) Üngör, B.; Köse, H.; Kurtulmaz, Yosum; Harmancı, A.
    We introduce a weakly symmetric ring which is a generalization of a symmetric ring and a strengthening of both a GWS ring and a weakly reversible ring, and investigate properties of the class of this kind of rings. A ring R is called weakly symmetric if for any a, b, c 2 R, abc being nilpotent implies that Racrb is a nil left ideal of R for each r 2 R. Examples are given to show that weakly symmetric rings need to be neither semicommutative nor symmetric. It is proved that the class of weakly symmetric rings lies also between those of 2-primal rings and directly finite rings. We show that for a nil ideal I of a ring R, R is weakly symmetric if and only if R=I is weakly symmetric. If R[x] is weakly symmetric, then R is weakly symmetric, and R[x] is weakly symmetric if and only if R[x; x-1] is weakly symmetric. We prove that a weakly symmetric ring which satises Köthe's conjecture is exactly an NI ring. We also deal with some extensions of weakly symmetric rings such as a Nagata extension, a Dorroh extension.
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    On π-Morphic modules
    (Hacettepe University, Department of Mathematics, 2013) Harmanci, A.; Kose, H.; Kurtulmaz, Yosum
    Let R be an arbitrary ring with identity and M be a right R-module with S = End(MR). Let f ∈ S. f is called π-morphic if M/f n(M) ∼=rM(fn) for some positive integer n. A module M is called π-morphic if every f ∈ S is π-morphic. It is proved that M is π-morphic and image-projective if and only if S is right π-morphic and M generates its kernel. S is unit-π-regular if and only if M is π-morphic and π-Rickart if and only if M is π-morphic and dual π-Rickart. M is π-morphic and image-injective if and only if S is left π-morphic and M cogenerates itscokernel.
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    Reflexivity of rings via nilpotent elements
    (Union Matematica Argentina, 2020) Harmancı, A.; Köse, H.; Kurtulmaz, Yosum; Üngör, B.
    An ideal I of a ring R is called left N-reflexive if for any a ∈ nil(R) and b ∈ R, aRb ⊆ I implies bRa ⊆ I, where nil(R) is the set of all nilpotent elements of R. The ring R is called left N-reflexive if the zero ideal is left N-reflexive. We study the properties of left N-reflexive rings and related concepts. Since reflexive rings and reduced rings are left N-reflexive rings, we investigate the sufficient conditions for left N-reflexive rings to be reflexive and reduced. We first consider basic extensions of left N-reflexive rings. For an ideal-symmetric ideal I of a ring R, R/I is left N-reflexive. If an ideal I of a ring R is reduced as a ring without identity and R/I is left N-reflexive, then R is left N-reflexive. If R is a quasi-Armendariz ring and the coefficients of any nilpotent polynomial in R[x] are nilpotent in R, it is proved that R is left N-reflexive if and only if R[x] is left N-reflexive. We show that the concept of left N-reflexivity is weaker than that of reflexivity and stronger than that of right idempotent reflexivity.
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    Rings having normality in terms of the Jacobson radical
    (Springer, 2020) Köse, H.; Kurtulmaz, Yosum; Harmancı, A.
    A ring R is defined to be J-normal if for any a,r∈Ra,r∈R and idempotent e∈Re∈R, ae=0ae=0 implies Rera⊆J(R)Rera⊆J(R), where J(R) is the Jacobson radical of R. The class of J-normal rings lies between the classes of weakly normal rings and left min-abel rings. It is proved that R is J-normal if and only if for any idempotent e∈Re∈R and for any r∈Rr∈R, R(1−e)re⊆J(R)R(1−e)re⊆J(R) if and only if for any n≥1n≥1, the n×nn×n upper triangular matrix ring Un(R)Un(R) is a J-normal ring if and only if the Dorroh extension of R by ZZ is J-normal. We show that R is strongly regular if and only if R is J-normal and von Neumann regular. For a J-normal ring R, it is obtained that R is clean if and only if R is exchange. We also investigate J-normality of certain subrings of the ring of 2×22×2 matrices over R.
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    Rings in which elements are a sum of a central and a unit element
    (The Belgian Mathematical Society, 2019) Kurtulmaz, Yosum; Halıcıoğlu, S.; Harmancı, A.; Chen, H.
    In this paper we introduce a new class of rings whose elements are a sum of a central and a unit element, namely a ring RR is called CUCU if each element a∈Ra∈R has a decomposition a=c+ua=c+u where cc is central and uu is unit. One of the main results in this paper is that if FF is a field which is not isomorphic to Z2Z2, then M2(F)M2(F) is a CUCU ring. This implies, in particular, that any square matrix over a field which is not isomorphic to Z2Z2 is the sum of a central matrix and a unit matrix.
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    Rings which are duo on Zhou radical
    (Springer, 2022-09-02) Ungor, B.; Harmanci, A.; Kurtulmaz, Yosum
    In ring theory, duoness and Zhou radical which is known as delta ideal have important roles. In this paper, we consider both concepts together by studying duoness on Zhou radical. By means of this study, we obtain a new kind of generalizations of commutativity. Firstly, we determine Zhou radical of some rings, then Zhou radical is applied to the duo property of rings, so we introduce a notion of right (left) dZr rings. We show that this notion is not left-right symmetric. We investigate some relations between right dZr rings and certain rings, and also deal with some ring extensions in terms of dZr property.
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    Semicommutativity of amalgamated rings
    (Journal of Mathematical Research with Applications, 2018) Köse, H.; Kurtulmaz, Yosum; Üngör, B.; Harmancı, A.
    In this paper, we study some cases when an amalgamated construction A ◃▹f I of a ring A along an ideal I of a ring B with respect to a ring homomorphism f from A to B, is prime, semiprime, semicommutative, nil-semicommutative and weakly semicommutative.
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    Structure theory of central simple ℤd-graded algebras
    (TÜBİTAK, 2012) Koç, C.; Kurtulmaz, Yosum
    This paper investigates the structure theory of ℤd- central simple graded algebras and gives the complete decomposition into building block algebras. The results are also applied to generalized Clifford algebras, which are motivating examples of ℤd-central simple graded algebras.