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Relevant bibliographies by topics / Duo-ring

Academic literature on the topic 'Duo-ring'

Author: Grafiati

Published: 30 May 2022

Last updated: 31 May 2022

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Journal articles on the topic "Duo-ring"

1

Harmanci, A., Y. Kurtulmaz, and B. Ungor. "Duo property for rings by the quasinilpotent perspective." Carpathian Mathematical Publications 13, no. 2 (October 17, 2021): 485–500. http://dx.doi.org/10.15330/cmp.13.2.485-500.

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In this paper, we focus on the duo ring property via quasinilpotent elements, which gives a new kind of generalizations of commutativity. We call this kind of rings qnil-duo. Firstly, some properties of quasinilpotents in a ring are provided. Then the set of quasinilpotents is applied to the duo property of rings, in this perspective, we introduce and study right (resp., left) qnil-duo rings. We show that this concept is not left-right symmetric. Among others, it is proved that if the Hurwitz series ring $H(R; \alpha)$ is right qnil-duo, then $R$ is right qnil-duo. Every right qnil-duo ring is abelian. A right qnil-duo exchange ring has stable range 1.

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Chatters, A. W., and Weimin Xue. "On right duo p.p. rings." Glasgow Mathematical Journal 32, no. 2 (May 1990): 221–25. http://dx.doi.org/10.1017/s0017089500009253.

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Throughout the paper, rings are associative rings with identity. A ring is called right duo if every right ideal is two-sided, and it is called right p.p. if every principal right ideal is projective. A left duo (p.p.) ring is denned similarly, and a duo (p.p.) ring will mean a ring which is both right and left duo (p.p.). There is a right p.p. ring that is not left p.p. (see Chase [2[). Small [9] proved that right p.p. implies left p.p. if there are no infinite sets of orthogonal idempotents, and Endo [5, Proposition 2] has shown the same implication in the case where each idempotent in the ring is central. Since Courter [3, Theorem 1.3] noted that every idempotent in a right duo ring is central, we can simply speak of right duo p.p. rings. A typical example of a right duo ring which is not left duo is the following. Let F be a field and F(x) the field of rational functions over F. Let R = F(x)× F(x) as an additive group and define the multiplication as follows:Then R is a local artinian ring with c(RR) = 2 and c(RR)= 3. Thus R is right duo but not left due.

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3

Dmytruk, A. A., A. I. Gatalevych, and M. I. Kuchma. "Stable range conditions for abelian and duo rings." Matematychni Studii 57, no. 1 (March 31, 2022): 92–97. http://dx.doi.org/10.30970/ms.57.1.92-97.

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The article deals with the following question: when does the classical ring of quotientsof a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are thereidempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regularrange 1, a ring of semihereditary range 1, a ring of regular range 1. We find relationshipsbetween the introduced classes of rings and known ones for abelian and duo rings.We proved that semihereditary local duo ring is a ring of semihereditary range 1. Also it was proved that a regular local Bezout duo ring is a ring of stable range 2. In particular, the following Theorem 1 is proved: For an abelian ring $R$ the following conditions are equivalent:$1.$\ $R$ is a ring of stable range 1; $2.$\ $R$ is a ring of von Neumann regular range 1. The paper also introduces the concept of the Gelfand element and a ring of the Gelfand range 1 for the case of a duo ring. Weproved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3).

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Bien, Mai Hoang, and Johan Öinert. "Quasi-duo differential polynomial rings." Journal of Algebra and Its Applications 17, no. 04 (April 2018): 1850072. http://dx.doi.org/10.1142/s021949881850072x.

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In this paper, we give a characterization of left (right) quasi-duo differential polynomial rings. In particular, we show that a differential polynomial ring is left quasi-duo if and only if it is right quasi-duo. This yields a partial answer to a question posed by Lam and Dugas in 2005. We provide nontrivial examples of such rings and give a complete description of the maximal ideals of an arbitrary quasi-duo differential polynomial ring. Moreover, we show that there is no left (right) quasi-duo differential polynomial ring in several indeterminates.

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5

Ghorbani, A., and M. Naji Esfahani. "On noncommutative FGC rings." Journal of Algebra and Its Applications 14, no. 07 (April 24, 2015): 1550109. http://dx.doi.org/10.1142/s0219498815501091.

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Many studies have been conducted to characterize commutative rings whose finitely generated modules are direct sums of cyclic modules (called FGC rings), however, the characterization of noncommutative FGC rings is still an open problem, even for duo rings. We study FGC rings in some special cases, it is shown that a local Noetherian ring R is FGC if and only if R is a principal ideal ring if and only if R is a uniserial ring, and if these assertions hold R is a duo ring. We characterize Noetherian duo FGC rings. In fact, it is shown that a duo ring R is a Noetherian left FGC ring if and only if R is a Noetherian right FGC ring, if and only if R is a principal ideal ring.

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6

Gao, Weidong, and Yuanlin Li. "On Duo Group Rings." Algebra Colloquium 18, no. 01 (March 2011): 163–70. http://dx.doi.org/10.1142/s1005386711000101.

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It is shown that if the group ring RQ8 of the quaternion group Q8 of order 8 over an integral domain R is duo, then R is a field for the following cases: (1) char R ≠ 0, and (2) char R = 0 and S ⊆ R ⊆ KS, where S is a ring of algebraic integers and KS is its quotient field. Hence, we confirm that the field ℚ of rational numbers is the smallest integral domain R of characteristic zero such that RQ8 is duo. A non-field integral domain R of characteristic zero for which RQ8 is duo is also identified. Moreover, we give a description of when the group ring RG of a torsion group G is duo.

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7

Hong, Chan Yong, Hong Kee Kim, Nam Kyun Kim, Tai Keun Kwak, and Yang Lee. "Duo Property on the Monoid of Regular Elements." Algebra Colloquium 29, no. 02 (April 30, 2022): 203–16. http://dx.doi.org/10.1142/s1005386722000165.

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We study the right duo property on regular elements, and we say that rings with this property are right DR. It is first shown that the right duo property is preserved by right quotient rings when the given rings are right DR. We prove that the polynomial ring over a ring [Formula: see text] is right DR if and only if [Formula: see text] is commutative. It is also proved that for a prime number [Formula: see text], the group ring [Formula: see text] of a finite [Formula: see text]-group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] is right DR if and only if it is right duo, and that there exists a group ring [Formula: see text] that is neither DR nor duo when [Formula: see text] is not a [Formula: see text]-group.

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8

Sorokin, O. S. "Finite homomorphic images of Bezout duo-domains." Carpathian Mathematical Publications 6, no. 2 (December 29, 2014): 360–66. http://dx.doi.org/10.15330/cmp.6.2.360-366.

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It is proved that for a quasi-duo Bezout ring of stable range 1 the duo-ring condition is equivalent to being an elementary divisor ring. As an application of this result a couple of useful properties are obtained for finite homomorphic images of Bezout duo-domains: they are coherent morphic rings, all injective modules over them are flat, their weak global dimension is either 0 or infinity. Moreover, we introduce the notion of square-free element in noncommutative case and it is shown that they are adequate elements of Bezout duo-domains. In addition, we are going to prove that these elements are elements of almost stable range 1, as well as necessary and sufficient conditions for being square-free element are found in terms of regularity, Jacobson semisimplicity, and boundness of weak global dimension of finite homomorphic images of Bezout duo-domains.

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9

Yu, Hua-Ping. "On quasi-duo rings." Glasgow Mathematical Journal 37, no. 1 (January 1995): 21–31. http://dx.doi.org/10.1017/s0017089500030342.

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Bass [1] proved that if R is a left perfect ring, then R contains no infinite sets of orthogonal idempotents and every nonzero left R-module has a maximal submodule, and asked if this property characterizes left perfect rings ([1], Remark (ii), p. 470). The fact that this is true for commutative rings was proved by Hamsher [12], and that this is not true in general was demonstrated by examples of Cozzens [7] and Koifman [14]. Hamsher's result for commutative rings has been extended to some noncommutative rings. Call a ring left duo if every left ideal is two-sided; Chandran [5] proved that Bass’ conjecture is true for left duo rings. Call a ring R weakly left duo if for every r ε R, there exists a natural number n(r) (depending on r) such that the principal left ideal Rrn(r) is two-sided. Recently, Xue [21] proved that Bass’ conjecture is still true for weakly left duo rings.

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10

Park, Chul-Hwan. "Intuitionistic fuzzy ideals in Regular duo ring." Journal of Korean Institute of Intelligent Systems 17, no. 1 (February 25, 2007): 112–17. http://dx.doi.org/10.5391/jkiis.2007.17.1.112.

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Books on the topic "Duo-ring"

1

Encyclopedia Duo/3 Ring Binder. 3rd ed. Pfeiffer & Co, 1991.

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Book chapters on the topic "Duo-ring"

1

Faye, Daouda, Mohamed Ben Fraj Ben Maaouia, and Mamadou Sanghare. "Localization in a Duo-Ring and Polynomials Algebra." In Non-Associative and Non-Commutative Algebra and Operator Theory, 183–91. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-32902-4_13.

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2

Bell, Howard E., and Yuanlin Li. "Reversible and Duo Group Rings." In Advances in Ring Theory, 37–46. Basel: Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-0346-0286-0_3.

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Conference papers on the topic "Duo-ring"

1

Bei Li, Muhammad Irfan Memon, Guohui Yuan, Zhuoran Wang, Siyuan Yu, Gabor Mezosi, and Marc Sorel. "All-optical response of semiconductor ring laser bistable to duo optical injections." In 2008 Conference on Lasers and Electro-Optics (CLEO). IEEE, 2008. http://dx.doi.org/10.1109/cleo.2008.4551764.

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