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Reversible rings with involutions and some minimalities
1Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.
Thescientificworldjournal
|February 4, 2014
Summary
This study introduces ∗-reversible rings, a new class of algebraic structures with involutions. We explore their properties and find that polynomial extensions may not preserve this property, with minimal examples identified.
Area of Science:
- Abstract algebra
- Ring theory
- Commutative algebra
Background:
- Recent advancements in extended reversibilities on rings.
- The established properties of reversible, symmetric, reflexive, and semicommutative rings.
Purpose of the Study:
- To introduce and investigate ∗-reversible rings, a novel class of rings featuring involutions.
- To explore fundamental properties and provide illustrative examples of ∗-reversible rings.
- To determine conditions under which polynomial rings derived from ∗-reversible rings may not exhibit the ∗-reversible property.
Main Methods:
- Theoretical investigation of algebraic structures.
- Development of criteria for identifying rings that cannot accommodate involutions.
- Analysis of ring properties including symmetry, reflexivity, and semicommutativity.
Main Results:
- Demonstration that polynomial rings of ∗-reversible rings are not necessarily ∗-reversible.
- Establishment of a criterion for rings lacking the capacity to adhere to any involution.
- Identification of a minimal noninvolutary ring of order 4.
- Identification of a minimal noncommutative ∗-reversible ring of order 16.
Conclusions:
- ∗-reversible rings represent a significant extension within the study of reversible rings.
- The behavior of polynomial extensions and the existence of minimal examples provide key insights into the structure of these rings.
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