Reflexive rings and their extensions

Rok, strany: 2013, 417 - 430

A right ideal $I$ is reflexive if $x R y \in I$ implies $y R x \in I$ for $x,y \in R$. We shall call a ring $R$ a reflexive ring if $a R b=0$ implies $b R a=0$ for $a, b\in R$. 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 $R[x]$ is reflexive if and only if $R[x; x^-1]$ 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 $R[x]$ is weakly reflexive.

ISO 690:

Zhao, L., Zhu, X., Gu, Q. 2013. Reflexive rings and their extensions. In Mathematica Slovaca, vol. 63, no.3, pp. 417-430. 0139-9918. DOI: https://doi.org/10.2478/s12175-013-0106-5

APA:

Zhao, L., Zhu, X., Gu, Q. (2013). Reflexive rings and their extensions. Mathematica Slovaca, 63(3), 417-430. 0139-9918. DOI: https://doi.org/10.2478/s12175-013-0106-5