Minimal Reflexive Nonsemicommutative Rings

TL;DR

This paper classifies minimal abelian reflexive nonsemicommutative rings, showing they are of order 256, and establishes that finite abelian reflexive rings of order p^k (k<8) are reversible, advancing ring taxonomy.

Contribution

It identifies and characterizes minimal abelian reflexive nonsemicommutative rings, including their order and properties, filling a gap in the taxonomy of 2-primal rings.

Findings

Minimal abelian reflexive nonsemicommutative rings have order 256.

All finite abelian reflexive rings of order p^k (k<8) are reversible.

Example of such a ring is F2D8.

Abstract

It has recently been shown that a minimal reversible nonsymmetric ring has order 256 answering a questioned original posed in a paper on a taxonomy of 2-primal rings. Answers to similar questions on minimal rings relating to this taxonomy were also answered in a related work. One type of minimal ring that was left out of that report, was a minimal abelian reflexive nonsemicommutative ring. In this work it is shown that a minimal abelian reflexive nonsemicommutative ring is of order 256 an example of which is F2D8. This is a consequence of the other primary result which is that a finite abelian reflexive ring of order pk for some prime p and k < 8 is reversible.