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Classifications of Several Classes of Armendariz-like Rings Relative to an Abelian Monoid and Its Applications

Author

Listed:
  • Jianwei He

    (School of Mathematics, Nanjing University, Nanjing 210093, China
    Current address: School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, China.
    These authors contributed equally to this work.)

  • Yajun Ma

    (School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China
    These authors contributed equally to this work.)

Abstract

Let M be an Abelian monoid. A necessary and sufficient condition for the class A r m M of all Armendariz rings relative to M to coincide with the class A r m of all Armendariz rings is given. As a consequence, we prove that A r m M has exactly three cases: the empty set, A r m , and the class of all rings. If N is an Abelian monoid, then we prove that A r m M × N = A r m M ⋂ A r m N , which gives a partial affirmative answer to the open question of Liu in 2005 (whether R is M × N -Armendariz if R is M -Armendariz and N -Armendariz). We also show that the other Armendariz-like rings relative to an Abelian monoid, such as M -quasi-Armendariz rings, skew M -Armendariz rings, weak M -Armendariz rings, M - π -Armendariz rings, nil M -Armendariz rings, upper nil M -Armendariz rings and lower nil M -Armendariz rings can be handled similarly. Some conclusions on these classes have, therefore, been generalized using these classifications.

Suggested Citation

  • Jianwei He & Yajun Ma, 2025. "Classifications of Several Classes of Armendariz-like Rings Relative to an Abelian Monoid and Its Applications," Mathematics, MDPI, vol. 13(5), pages 1-22, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:874-:d:1606224
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