Minimal reversible nonsymmetric rings

Abstract

Marks showed that F2Q8, the F2 group algebra over the quaternion group, is a reversible nonsymmetric ring, then questioned whether or not this ring is minimal with respect to cardinality. In this work, it is shown that the cardinality of a minimal reversible nonsymmetric ring is indeed 256. Furthermore, it is shown that although F2Q8 is a duo ring, there are also examples of minimal reversible nonsymmetric rings which are nonduo.

Some minimal rings related to 2-primal rings

“…Is F 2 Q 8 a minimal reversible nonsymmetric ring? In [15], the present author shows that F 2 Q 8 is indeed such a ring. It is also shown that F 2 u, v u 3 , v 3 , u 2 + v 2 + vu, vu 2 + uvu + vuv is a minimal reversible nonsymmetric ring as well.…”

“…By Lemma 2.7, R is reversible. In [15] it was shown that a minimal reversible nonsymmetric ring is of order 256. Since |R| = 16, R is symmetric.…”

“…and R is reversible. From [15], a reversible ring of order less than 256 is symmetric. Since |R| = 32, R is symmetric.…”

Minimal Reflexive Nonsemicommutative Rings

“…Even more, F 2 Q 8 and F 2 D 8 are examples of the rings just mentioned which are the only two noncommutative group algebras over F 2 . The connection between these two ring classes will be clear when the proofs to the present results are compared to the results in [14]. The important connection is that a ring is reversible if and only if it is reflexive and semicommutative, a fact which is exploited to obtain the current results.…”

“…A taxonomy of rings related to 2-primal rings was given in [11] where the question was asked whether or not F 2 Q 8 was a minimal reversible nonsymmetric ring. This was answered positively in [14]. Another such example was also given there, namely…”