Reflexive property on rings with involution

2019 ◽  
Vol 12 (07) ◽  
pp. 2050011
Author(s):  
Arnab Bhattacharjee ◽  
Uday Shankar Chakraborty
Keyword(s):  
Ring Theory ◽  
Trivial Extension ◽  
Noncommutative Ring ◽  
Rings With Involution ◽  
Noncommutative Ring Theory

Mason introduced the notion of reflexive property for rings which play roles in noncommutative ring theory. In this paper, we extend this property to rings with involution and investigate their properties. We provide many examples of these rings and also consider some extensions such as trivial extension, Dorroh extension, etc.

Insertion of units at zero products

2018 ◽  
Vol 17 (03) ◽  
pp. 1850043 ◽  
Author(s):  
Hong Kee Kim ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Yeonsook Seo
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Ring Theory ◽  
Noncommutative Ring ◽  
Matrix Rings ◽  
Zero Divisors ◽  
Noncommutative Ring Theory ◽  
Equivalent Conditions

The purpose of this paper is to provide useful connections between units and zero divisors, by investigating the structure of a class of rings in which Köthe’s conjecture (i.e. the sum of two nil left ideals is nil) holds. We introduce the concept of unit-IFP for the purpose, in relation with the inserting property of units at zero products. We first study the relation between unit-IFP rings and related ring properties in a kind of matrix rings which has roles in noncommutative ring theory. The Jacobson radical of the polynomial ring over a unit-IFP ring is shown to be nil. We also provide equivalent conditions to the commutativity via the unit-IFP of such matrix rings. We construct examples and counterexamples which are necessary to the naturally raised questions.

scholarly journals A Survey on Some Algebraic Characterizations of Hilbert’s Nullstellensatz for Non-commutative Rings of Polynomial Type

2020 ◽  
Vol 16 (31) ◽  
pp. 27-52
Author(s):  
Armando Reyes ◽  
Jason Hernández-Mogollón
Keyword(s):  
Algebraic Geometry ◽  
Commutative Rings ◽  
Ring Theory ◽  
Noncommutative Ring ◽  
Polynomial Type ◽  
Noncommutative Ring Theory

In this paper we present a survey of some algebraic characterizations of Hilbert’s Nullstellensatz for non-commutative rings of polynomial type. Using several results established in the literature, we obtain a version of this theorem for the skew Poincaré-Birkhoff-Witt extensions. Once this is done, we illustrate the Nullstellensatz with examples appearing in noncommutative ring theory and non-commutative algebraic geometry.

Structure of insertion property by powers

2018 ◽  
Vol 28 (03) ◽  
pp. 501-519
Author(s):  
Jeoung Soo Cheon ◽  
Hai-Lan Jin ◽  
Da Woon Jung ◽  
Hong Kee Kim ◽  
Yang Lee ◽  
...  
Keyword(s):  
Ring Theory ◽  
Polynomial Rings ◽  
Noncommutative Ring ◽  
Finite Rings ◽  
Matrix Rings ◽  
Noncommutative Ring Theory

This paper, concerns a class of rings which satisfies the Abelian property in relation to the insertion property at zero by powers and local finite. The concepts of Insertion of-Power-Factors-Property (PFP) and principal finite are introduced for the purpose, and the structures of IPFP, Abelian and locally (principally) finite rings are investigated in relation with several situations of matrix rings and polynomial rings. Moreover, the results obtained here are widely applied to various sorts of rings which have roles in the noncommutative ring theory.

scholarly journals A Characterization of Derivations in Prime Rings with Involution

2019 ◽  
Vol 12 (3) ◽  
pp. 1138-1148
Author(s):  
Shakir Ali ◽  
M. Rahman Mozumder ◽  
Adnan Abbasi ◽  
M. Salahuddin Khan
Keyword(s):  
Noncommutative Ring ◽  
Prime Rings ◽  
Differential Identities ◽  
Rings With Involution ◽  
Torsion Free

The purpose of this paper is to investigate $*$-differential identities satisfied by pair of derivations on prime rings with involution. In particular, we prove that if a 2-torsion free noncommutative ring $R$ admit nonzero derivations $d_1, d_2$ such that $[d_1(x), d_2(x^*)]=0$ for all $x\in R$, then $d_1=\lambda d_2$, where $\lambda\in C$. Finally, we provide an example to show that the condition imposed in the hypothesis of our results are necessary.

A new class of non-semiprime quasi-Armendariz rings

2014 ◽  
Vol 51 (2) ◽  
pp. 165-171
Author(s):  
Mohammad Habibi
Keyword(s):  
Polynomial Ring ◽  
Trivial Extension ◽  
Sufficient Condition ◽  
Nilpotent Ideal ◽  
Noncommutative Ring ◽  
Polynomial Extension ◽  
New Class ◽  
Monoid Ring ◽  
Armendariz Rings

Hirano [On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168 (2002), 45–52] studied relations between the set of annihilators in a ring R and the set of annihilators in a polynomial extension R[x] and introduced quasi-Armendariz rings. In this paper, we give a sufficient condition for a ring R and a monoid M such that the monoid ring R[M] is quasi-Armendariz. As a consequence we show that if R is a right APP-ring, then R[x]=(xn) and hence the trivial extension T(R,R) are quasi-Armendariz. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which are quasi-Armendariz.

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