Essentially quasi-duo rings

• Type: Another Publication in an International Scientific Journal
• Date: 2025

Authors

Christian Lomp (Author) FCUP; Yousif, M (Author) Other; Zhou, Y (Author) Other

Journal

Title: Contemporary Mathematics

Vol. 24

Pages: 161-174

ISSN: 0271-4132

Publisher: American Mathematical Society

Scientific classification

CORDIS: Physical sciences > Mathematics > Algebra

FOS: Natural sciences > Mathematics

Associated Projects

MAT - Financiamento Plurianual da Unidade de I&D CMUP - Centro de Investigação em Matemática da Universidade do Porto

Other information

Authenticus ID: P-01A-V4B

DOI: 10.1090/conm/826/16589

Abstract (EN): <p>A ring with identity is called essentially right quasi-duo if every essential maximal right ideal of it is a two-sided ideal. Essentially right quasi-duo rings generalize essentially right duo rings, a notion that arose in the study of hypercyclic rings, and right quasi-duo rings, as introduced by S.H. Brown. We prove that a ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is essentially right quasi-duo if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is semisimple or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R slash upper S o c left-parenthesis upper R Subscript upper R Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>S</mml:mi> <mml:mi>o</mml:mi> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>R</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R/Soc(R_R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is right quasi-duo. Although it is still unknown, whether a right quasi-duo ring is left quasi-duo, we provide an example of an essentially right quasi-duo ring that is not essentially left quasi-duo. Furthermore, while exchange right quasi-duo rings are known to be clean, there exist exchange essentially right quasi-duo rings that are not clean. A thorough study of essentially right quasi-duo rings is carried out and their relationship to skew power series rings, trivial extensions and formal triangular matrix rings is explored.</p>

Language: English

Type (Professor's evaluation): Scientific