Browsing by Author "Köse, H."

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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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    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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    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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    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.