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SYMMETRICITY AND REVERSIBILITY FROM THE PERSPECTIVE OF NILPOTENTS
Harmanci, Abdullah,Kose, Handan,Ungor, Burcu Korean Mathematical Society 2021 대한수학회논문집 Vol.36 No.2
In this paper, we deal with the question that what kind of properties does a ring gain when it satisfies symmetricity or reversibility by the way of nilpotent elements? By the motivation of this question, we approach to symmetric and reversible property of rings via nilpotents. For symmetricity, we call a ring R middle right-(resp. left-)nil symmetric (mr-nil (resp. ml-nil) symmetric, for short) if abc = 0 implies acb = 0 (resp. bac = 0) for a, c ∈ R and b ∈ nil(R) where nil(R) is the set of all nilpotent elements of R. It is proved that mr-nil symmetric rings are abelian and so directly finite. We show that the class of mr-nil symmetric rings strictly lies between the classes of symmetric rings and weak right nil-symmetric rings. For reversibility, we introduce left (resp. right) N-reversible ideal I of a ring R if for any a ∈ nil(R), b ∈ R, being ab ∈ I implies ba ∈ I (resp. b ∈ nil(R), a ∈ R, being ab ∈ I implies ba ∈ I). A ring R is called left (resp. right) N-reversible if the zero ideal is left (resp. right) N-reversible. Left N-reversibility is a generalization of mr-nil symmetricity. We exactly determine the place of the class of left N-reversible rings which is placed between the classes of reversible rings and CNZ rings. We also obtain that every left N-reversible ring is nil-Armendariz. It is observed that the polynomial ring over a left N-reversible Armendariz ring is also left N-reversible.
On Quasi-Baer and p.q.-Baer Modules
Basser, Muhittin,Harmanci, Abdullah Department of Mathematics 2009 Kyungpook mathematical journal Vol.49 No.2
For an endomorphism ${\alpha}$ of R, in [1], a module $M_R$ is called ${\alpha}$-compatible if, for any $m{\in}M$ and $a{\in}R$, ma = 0 iff $m{\alpha}(a)$ = 0, which are a generalization of ${\alpha}$-reduced modules. We study on the relationship between the quasi-Baerness and p.q.-Baer property of a module MR and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of [2] and some results in [9]. In particular, we show: for an ${\alpha}$-compatible module $M_R$ (1) $M_R$ is p.q.-Baer module iff $M[x;{\alpha}]_{R[x;{\alpha}]}$ is p.q.-Baer module. (2) for an automorphism ${\alpha}$ of R, $M_R$ is p.q.-Baer module iff $M[x,x^{-1};{\alpha}]_{R[x,x^{-1};{\alpha}]}$ is p.q.-Baer module.
GENERALIZED SEMI COMMUTATIVE RINGS AND THEIR EXTENSIONS
Baser, Muhittin,Harmanci, Abdullah,Kwak, Tai-Keun Korean Mathematical Society 2008 대한수학회보 Vol.45 No.2
For an endomorphism ${\alpha}$ of a ring R, the endomorphism ${\alpha}$ is called semicommutative if ab=0 implies $aR{\alpha}(b)$=0 for a ${\in}$ R. A ring R is called ${\alpha}$-semicommutative if there exists a semicommutative endomorphism ${\alpha}$ of R. In this paper, various results of semicommutative rings are extended to ${\alpha}$-semicommutative rings. In addition, we introduce the notion of an ${\alpha}$-skew power series Armendariz ring which is an extension of Armendariz property in a ring R by considering the polynomials in the skew power series ring $R[[x;\;{\alpha}]]$. We show that a number of interesting properties of a ring R transfer to its the skew power series ring $R[[x;\;{\alpha}]]$ and vice-versa such as the Baer property and the p.p.-property, when R is ${\alpha}$-skew power series Armendariz. Several known results relating to ${\alpha}$-rigid rings can be obtained as corollaries of our results.
On Semicommutative Modues and Rings
AGAYEV, NAZIM,HARMANCI, ABDULLAH 대한수학회 2007 Kyungpook mathematical journal Vol.47 No.1
We say a module M_(R) a semicommutative module if for any m ∈ M and any a ∈ R, ma = 0 implies mRa = 0. This paper gives various properties of reduced, Armendariz, Baer, Quasi-Baer, p.p. and p.q.-Baer rings to extend to modules. In addition we also prove, for a p.p.-ring R, R is semicommutative iff R is Armendariz. Let R be an abelian ring and M_(R) be a pp.-module, then MR is a semicommutative module iff M_(R) is an Armendariz module. For any ring R, R is semicommutative iff A(R, a) is semicommutative. Let R be a reduced ring, it is shown that for number n ≥ 4 and k = [n/2], T_(n)^(k)(R) is semicommutative ring but T_(n)^(k-1)(R) is not.
On a Class of Semicommutative Rings
Ozen, Tahire,Agayev, Nazim,Harmanci, Abdullah Department of Mathematics 2011 Kyungpook mathematical journal Vol.51 No.3
In this paper, a generalization of the class of semicommutative rings is investigated. A ring R is called central semicommutative if for any a, b ${\in}$ R, ab = 0 implies arb is a central element of R for each r ${\in}$ R. We prove that some results on semicommutative rings can be extended to central semicommutative rings for this general settings.
Modules Which Are Lifting Relative To Module Classes
Kosan, Muhammet Tamer,Harmanci, Abdullah Department of Mathematics 2008 Kyungpook mathematical journal Vol.48 No.1
In this paper, we study a module which is lifting and supplemented relative to a module class. Let R be a ring, and let X be a class of R-modules. We will define X-lifting modules and X-supplemented modules. Several properties of these modules are proved. We also obtain results for the case of specific classes of modules.
Generalized semicommutative rings and their extensions
Muhittin Baser,Abdullah Harmanci,곽태근 대한수학회 2008 대한수학회보 Vol.45 No.2
For an endomorphism α of a ring R, the endomorphism α is called semicommutative if ab = 0 implies aRα (b) = 0 for a ∈ R. A ring R is called α-semicommutative if there exists a semicommutative endomor-phism α of R. In this paper, various results of semicommutative ringsare extended to α-semicommutative rings. In addition, we introduce thenotion of an α-skew power series Armendariz ring which is an extensionof Armendariz property in a ring R by considering the polynomials in theskew power series ring R[[x;α]]. We show that a number of interestingproperties of a ringR transfer to its the skew power series ringR[[x;α]]and vice-versa such as the Baer property and the p.p.-property, when R is α-skew power series Armendariz. Several known results relating to α-rigid rings can be obtained as corollaries of our results.
Strong J-cleanness of formal matrix rings
O. Gurgun,S. Halicioglu,A. Harmanci 장전수학회 2014 Advanced Studies in Contemporary Mathematics Vol.24 No.4
An element a of a ring R is called strongly J-clean provided that there exists an idempotent e∈R such that a-e∈J(R) and ae = ea. A ring R is strongly J-clean in case every element in R is strongly J-clean. In the paper, we investigate strong J-cleanness of M2(R;S) for a local ring R and s∈R. We determine the conditions under which elements of M2(R;s) are strongly J-clean.
SYMMETRIC PROPERTY OF RINGS WITH RESPECT TO THE JACOBSON RADICAL
Calci, Tugce Pekacar,Halicioglu, Sait,Harmanci, Abdullah Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.1
Let R be a ring with identity and J(R) denote the Jacobson radical of R, i.e., the intersection of all maximal left ideals of R. A ring R is called J-symmetric if for any $a,b,c{\in}R$, abc = 0 implies $bac{\in}J(R)$. We prove that some results of symmetric rings can be extended to the J-symmetric rings for this general setting. We give many characterizations of such rings. We show that the class of J-symmetric rings lies strictly between the class of symmetric rings and the class of directly finite rings.