This paper defines an isomorphism, an anti-isomorphism and a Jordan isomorphism in a gamma ring and develops some important results relating to these concepts. Using these results we prove Herstein’s theorem of classical rings in case of prime gamma rings by showing that every Jordan isomorphism of a 2-torsion free prime gamma ring is either an isomorphism or an anti-isomorphism. AMS Mathematic...

It is examined in several papers when is it true the isomorphism of two cylindric set algebras is a base-isomorphism. Problem 1. For which M ∈ CAα it is true that if for i = 0 and i = 1 Di ∈ Caα, f i : M→Di is an isomorphisms, and base Di = Vi imply that there exists a bijection g : V0 → V1 such that g = f1 ◦ f −1 0 (for notations and notions see [3] and [5]). By the Löwenheim-Skolem theorem th...

The division algebra of real quaternions, as the only noncommutative normed division real algebra up to isomorphism of normed algebras, is of great importance. In this note, first we present a brief introduction to quaternion matrices and quaternion linear algebra. This, among other things, will help us present the counterpart of a theorem of Herman Auerbach in the setting of quaternions. More ...

This submission contains theories that lead to a formalization of the proof of the Jordan-Hölder theorem about composition series of finite groups. The theories formalize the notions of isomorphism classes of groups, simple groups, normal series, composition series, maximal normal subgroups. Furthermore, they provide proofs of the second isomorphism theorem for groups, the characterization theo...

Abstract We prove a strengthened version of Shavrukov’s result on the non-isomorphism diagonalizable algebras two $\Sigma _1$ -sound theories, based improvements previously found by Adamsson. then obtain several corollaries to applying it various pairs theories and new examples. In particular, we show that there are no surjective homomorphisms from algebra $(\mathfrak {L}_T, \Box _T\Box _T)$ on...