Reflexive rings and their extensions |
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Authors: | Liang Zhao Xiaosheng Zhu Qinqin Gu |
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Institution: | 1106. School of Mathematics and Physics, Anhui University of Technology, Maanshan, 243032, P. R. China 2106. Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China
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Abstract: | A right ideal I is reflexive if xRy ∈ I implies yRx ∈ I for x, y ∈ R. We shall call a ring R a reflexive ring if aRb = 0 implies bRa = 0 for a, b ∈ R. We study the properties of reflexive rings and related concepts. We first consider basic extensions of reflexive rings. For a reduced iedal I of a ring R, if R/I is reflexive, we show that R is reflexive. We next discuss the reflexivity of some kinds of polynomial rings. For a quasi-Armendariz ring R, it is proved that R is reflexive if and only if Rx] is reflexive if and only if Rx; x ?1] is reflexive. For a right Ore ring R with Q its classical right quotient ring, we show that if R is a reflexive ring then Q is also reflexive. Moreover, we characterize weakly reflexive rings which is a weak form of reflexive rings and investigate its properties. Examples are given to show that weakly reflexive rings need not be semicommutative. It is shown that if R is a semicommutative ring, then Rx] is weakly reflexive. |
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