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Unified Extensions of Strongly Reversible Rings and Links with Other Classic Ring Theoretic Properties


Affiliations
1 Indian Institute of Technology, New Delhi-110016, India
2 Hamdard University, New Delhi 110 062, India
     

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Let R be a ring, (M, ≤) a strictly ordered monoid and ω : M → End(R) a monoid homomorphism. The skew generalized power series ring R[[M; ω]] is a compact generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomials rings, (skew) Laurent power series rings, (skew) group rings, (skew) monoid rings, Mal'cev Neumann rings and generalized power series rings. In this paper, we introduce concept of strongly (M, ω)-reversible ring (strongly reversible ring related to skew generalized power series ring R[[M, ω]]) which is a uni ed generalization of strongly reversible ring and study basic properties of strongly (M; ω)-reversible. The Nagata extension of strongly reversible is proved to be strongly reversible if R is Armendariz. Finally, it is proved that strongly reversible ring strictly lies between reduced and reversible ring in the expanded diagram given by Diesl et. al. [7].

Keywords

Reduced Ring, Armendariz Ring, Reversible Ring, Linear Armendariz Ring, Symmetric Ring, Duo Ring, Semi-Commutative Ring, Strongly Reversible Ring, Strongly (M; ω)-Reversible Ring.
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  • Unified Extensions of Strongly Reversible Rings and Links with Other Classic Ring Theoretic Properties

Abstract Views: 277  |  PDF Views: 1

Authors

R. K. Sharma
Indian Institute of Technology, New Delhi-110016, India
Amit B. Singh
Hamdard University, New Delhi 110 062, India

Abstract


Let R be a ring, (M, ≤) a strictly ordered monoid and ω : M → End(R) a monoid homomorphism. The skew generalized power series ring R[[M; ω]] is a compact generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomials rings, (skew) Laurent power series rings, (skew) group rings, (skew) monoid rings, Mal'cev Neumann rings and generalized power series rings. In this paper, we introduce concept of strongly (M, ω)-reversible ring (strongly reversible ring related to skew generalized power series ring R[[M, ω]]) which is a uni ed generalization of strongly reversible ring and study basic properties of strongly (M; ω)-reversible. The Nagata extension of strongly reversible is proved to be strongly reversible if R is Armendariz. Finally, it is proved that strongly reversible ring strictly lies between reduced and reversible ring in the expanded diagram given by Diesl et. al. [7].

Keywords


Reduced Ring, Armendariz Ring, Reversible Ring, Linear Armendariz Ring, Symmetric Ring, Duo Ring, Semi-Commutative Ring, Strongly Reversible Ring, Strongly (M; ω)-Reversible Ring.

References