• Title/Summary/Keyword: hereditary class

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LOWER AND UPPER FORMATION RADICAL OF NEAR-RINGS

  • Saxena, P.K.;Bhandari, M.C.
    • Kyungpook Mathematical Journal
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    • v.19 no.2
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    • pp.205-211
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    • 1979
  • In this paper we continue the study of formation radical (F-radical) classes initiated in [3]. Hereditary and stronger properties of F-radical classes are discussed by giving construction for lower hereditary, lower stronger and lower strongly hereditary F-radical classes containing a given class M. It is shown that the Baer F-radical B is the lower strongly hereditary F-radical class containing the class of all nilpotent ideals and it is the upper radical class with $\{(I,\;N){\mid}N{\in}C,\;N\;is\;prime\}{\subset}SB$ where SB denotes the semisimple F-radical class of B and C is an arbitrary but fixed class of homomorphically closed near-rings. The existence of a largest F-radical class contained in a given class is examined using the concept of complementary F-radical introduced by Scott [5].

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ON STRONGLY GORENSTEIN HEREDITARY RINGS

  • Hu, Kui;Kim, Hwankoo;Wang, Fanggui;Xu, Longyu;Zhou, Dechuan
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.2
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    • pp.373-382
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    • 2019
  • In this note, we mainly discuss strongly Gorenstein hereditary rings. We prove that for any ring, the class of SG-projective modules and the class of G-projective modules coincide if and only if the class of SG-projective modules is closed under extension. From this we get that a ring is an SG-hereditary ring if and only if every ideal is G-projective and the class of SG-projective modules is closed under extension. We also give some examples of domains whose ideals are SG-projective.

ON 𝜃-MODIFICATIONS OF GENERALIZED TOPOLOGIES VIA HEREDITARY CLASSES

  • Al-Omari, Ahmad;Modak, Shyamapada;Noiri, Takashi
    • Communications of the Korean Mathematical Society
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    • v.31 no.4
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    • pp.857-868
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    • 2016
  • Let (X, ${\mu}$) be a generalized topological space (GTS) and $\mathcal{H}$ be a hereditary class on X due to $Cs{\acute{a}}sz{\acute{a}}r$ [8]. In this paper, we define an operator $()^{\circ}:\mathcal{P}(X){\rightarrow}\mathcal{P}(X)$. By setting $c^{\circ}(A)=A{\cup}A^{\circ}$ for every subset A of X, we define the family ${\mu}^{\circ}=\{M{\subseteq}X:X-M=c^{\circ}(X-M)\}$ and show that ${\mu}^{\circ}$ is a GT on X such that ${\mu}({\theta}){\subseteq}{\mu}^{\circ}{\subseteq}{\mu}^*$, where ${\mu}^*$ is a GT in [8]. Moreover, we define and investigate ${\mu}^{\circ}$-codense and strongly ${\mu}^{\circ}$-codense hereditary classes.

On n-Amitsur Rings

  • Ochirbat, Baatar;Mendes, Deolinda I.C.;Tumurbat, Sodnomkhorloo
    • Kyungpook Mathematical Journal
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    • v.60 no.4
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    • pp.711-721
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    • 2020
  • The concepts of an Amitsur ring and a hereditary Amitsur ring, which were introduced and studied by S. Tumurbat in a recent paper, are generalized. For a positive integer n, a ring A is said to be an n-Amitsur ring if γ(A[Xn]) = (γ(A[Xn]) ∩ A)[Xn] for all radicals γ, where A[Xn] is the polynomial ring over A in n commuting indeterminates. If a ring A satisfies the above equation for all hereditary radicals γ, then A is said to be a hereditary n-Amitsur ring. Characterizations and examples of these rings are provided. Moreover, new radicals associated with n-Amitsur rings are introduced and studied. One of these is a special radical and its semisimple class is polynomially extensible.

SOME RESULTS ON 2-STRONGLY GORENSTEIN PROJECTIVE MODULES AND RELATED RINGS

  • Dong Chen;Kui Hu
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.4
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    • pp.895-903
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    • 2023
  • In this paper, we give some results on 2-strongly Gorenstein projective modules and related rings. We first investigate the relationship between strongly Gorenstein projective modules and periodic modules and then give the structure of modules over strongly Gorenstein semisimple rings. Furthermore, we prove that a ring R is 2-strongly Gorenstein hereditary if and only if every ideal of R is Gorenstein projective and the class of 2-strongly Gorenstein projective modules is closed under extensions. Finally, we study the relationship between 2-Gorenstein projective hereditary and 2-Gorenstein projective semisimple rings, and we also give an example to show the quotient ring of a 2-Gorenstein projective hereditary ring is not necessarily 2-Gorenstein projective semisimple.

SHARP HEREDITARY CONVEX RADIUS OF CONVEX HARMONIC MAPPINGS UNDER AN INTEGRAL OPERATOR

  • Cheny, Xingdi;Mu, Jingjing
    • Korean Journal of Mathematics
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    • v.24 no.3
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    • pp.369-374
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    • 2016
  • In this paper, we study the hereditary convex radius of convex harmonic mapping $f(z)=f_1(z)+{\bar{f_x(z)}}$ under the integral operator $I_f(z)={\int_{o}^{z}}{\frac{f_1(u)}{u}}du+{\bar{{\int_{o}^{z}}{\frac{f_x(u)}{u}}}}$ and obtain the sharp constant ${\frac{{\sqrt[4]{6}}-{\sqrt[]{15}}}{9}}$, which generalized the result corresponding to the class of analytic functions given by Nash.

Constitutive Model of Tendon Responses to Multiple Cyclic Demands (II) -Theory and Comparison-

  • Chun, Keyoung-Jin;Robert P. Hubbard
    • Journal of Mechanical Science and Technology
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    • v.15 no.9
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    • pp.1281-1291
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    • 2001
  • The hereditary integral form of a quasi-linear viscoelastic law has been employed. Four new concepts have been employed: 1. a reduced relaxation function with a non-linear exponential function of time, 2. an inverse method to determine the scale factor of the elastic response, 3. an instant elastic recovery strain during unloading, and 4. the results of a constitutive model for cyclic tests may be a function of the Heavyside class. These concepts have been supported by agreement between measured and predicted responses of soft connective tissue to three types of multiple cyclic tests which include rest periods of no extension and alternations between different strain levels. Such agreement has not been attained in the previous studies. Chun and Hubbard (2001) is our companion experimental analysis paper.

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LOWER FORMATION RADICAL FOR NEAR RINGS

  • Saxena, P.K.;Bhandari, M.C.
    • Kyungpook Mathematical Journal
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    • v.18 no.1
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    • pp.23-29
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    • 1978
  • In [7) Scott has defined C-formation radical for a class C of near rings and has studied its porperties under chain conditions. A natural question that arises is: Does there exist a Lower C-Formation radical class L(M) containing a given class M of ideals of near rings in C? In this paper we answer this by giving. two constructions for L(M) and prove that prime radical is hereditary.

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