On the Jacobson radical of strongly group graded rings

summary:For any non-torsion group $G$ with identity $e$, we construct a strongly $G$-graded ring $R$ such that the Jacobson radical $J(R_e)$ is locally nilpotent, but $J(R)$ is not locally nilpotent. This answers a question posed by Puczy{\l}owski

Optimization and matrix constructions for classification of data

Max-plus algebras and more general semirings have many useful applications and have been actively investigated. On the other hand, structural matrix rings are also well known and have been considered by many authors. The main theorem of this article completely describes all optimal ideals in the more general structural matrix semirings. Originally, our investigation of these ideals was motivated by applications in data mining for the design of centroid-based classification systems, as well as for the design of multiple classification systems combining several individual classifiers

The automorphisms of Petit's algebras

Let σ be an automorphism of a field K with fixed field F. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras K[t; σ]=fK[t; σ] obtained when the twisted polynomialf 2 K[t; σ] is invariant, and were first defined by Petit. We compute all their automorphisms if V commutes with all automorphisms in AutF (K) and n < m-1. In thecase where K=F is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over F. We also briefly investigate when two such algebras are isomorphic

Radicals Commuting With Bands Of Semigroups

F25.72&gt; y) 2 ae(S) =) (f(x); f(y)) 2 ae(T ) for any homomorphism f : S ! T ; (R2) ae(S=ae(S)) = &apos; for any s 2 A. Throughout, we shall consider radicals on the class of all semigroups. The congruence ae(S) is called the ae-radical of S. A semigroup S is said to be ae-semisimple (ae-radical) if ae(S) = &apos; (or ae(S) = !). If it is clear which radical is under consideration, then we shall omit the prefix `ae-&apos;. The class&lt