We consider the incidence semirings of graphs and prove that every incidence semiring has convenient visible bases for its right ideals and for its left ideals, and that these visible bases can be used to determine the weights of all right ideals that have maximum weight and all left ideals that have maximum weight.

We introduce a new construction involving Rees matrix semigroups and max-plus algebras that is very convenient for generating sets of centroids. We describe completely all optimal sets of centroids for all Rees matrix semigroups without any restrictions on the sandwich matrices.

The max-plus algebra is well known and has useful applications in the investigation of discrete event systems and affine equations. Structural matrix rings have been considered by many authors too. This article introduces more general structural matrix semirings, which include all matrix semirings over the max-plus algebra. We investigate properties of ideals in this construction motivated by applications to the design of centroid-based classification systems, or classifiers, as well as multiple classifiers combining several initial classifiers. The first main theorem of this paper shows that structural matrix semirings possess convenient visible generating sets for ideals. Our second main theorem uses two special sets to determine the weights of all ideals and describe all matrix ideals with the largest possible weight, which are optimal for the design of classification systems.

This paper continues the investigation of semigroup constructions motivated by applications in data mining. We give a complete description of the error-correcting capabilities of a large family of clusterers based on Rees matrix semigroups well known in semigroup theory. This result strengthens and complements previous formulas recently obtained in the literature. Examples show that our theorems do not generalize to other classes of semigroups.

Drensky and Lakatos (Lecture Notes in Computer Science, 357 (Springer, Berlin, 1989), pp. 181–188) have established a convenient property of certain ideals in polynomial quotient rings, which can now be used to determine error-correcting capabilities of combined multiple classifiers following a standard approach explained in the well-known monograph by Witten and Frank (Data Mining: Practical Machine Learning Tools and Techniques (Elsevier, Amsterdam, 2005)). We strengthen and generalise the result of Drensky and Lakatos by demonstrating that the corresponding nice property remains valid in a much larger variety of constructions and applies to more general types of ideals. Examples show that our theorems do not extend to larger classes of ring constructions and cannot be simplified or generalised.

For all finite p-groups, we describe the Jacobson radicals of incidence rings of group automata.

Giving as answer to Bergman's question, Cohen and Montgomery proved that, for every finite group G with identity e and each G-graded ring R = ⊕g∈GRg, the Jacobson radical J(Re) of the initial component Re is equal to Re ∩ J(R). We describe all semigroups S, which satisfy the following natural analogue of this property: J(Re) = Re ∩ J(R) for each S-graded ring R = ⊕s∈SRs and every idempotent e ∈ S.

Matrices provide essential tools in many branches of mathematics, and matrix semigroups have applications in various areas. In this paper we give a complete description of all infinite matrix semigroups satisfying a certain combinatorial property defined in terms of power graphs.

We describe all directed graphs with graph algebras isomorphic to syntactic semigroups of languages.

Suppose that a ring is a sum of its nilpotent subrings. We use directed graphs to give new conditions sufficient for the whole ring to be nilpotent.

We construct a ring $R$ which is a sum of two subrings $A$ and $B$ such that the Levitzki radical of $R$ does not contain any of the hyperannihilators of $A$ and $B$ . This answers an open question asked by Kegel in 1964.

For any ring R graded by a finite group, we give a bound on the classical Krull dimension of R in terms of the dimension of the initial component Re. It follows that if Re has finite classical Krull dimension, then the same is true of the whole ring R, too.

We describe all pairs of semigroups S and radicals ρ, such that ρ is invariant in S-graded rings. This generalises several known results due to Amitsur and Sands.

We give necessary and sufficient conditions on the semigroup S for the Jacobson radical to be S-invariant.

A number of classical theorems of ring theory deal with nilness and nilpotency of the Jacobson radical of various ring constructions (see [10], [18]). Several interesting results of this sort have appeared in the literature recently. In particular, it was proved in [1] that the Jacobson radical of every finitely generated PI-ring is nilpotent. For every commutative semigroup ring RS, it was shown in [11] that if J(R) is nil then J(RS) is nil. This result was generalized to all semigroup algebras satisfying polynomial identities in [15] (see [16, Chapter 21]). Further, it was proved in [12] that, for every normal band B, if J(R) is nilpotent, then J(RB) is nilpotent. A similar result for special band-graded rings was established in [13, Section 6]. Analogous theorems concerning nilpotency and local nilpotency were proved in [2] for rings graded by finite and locally finite semigroups.

Band sums of associative rings were introduced by Weissglass in 1973. The main theorem claims that the support of every Artinian band sum of rings is finite. This result is analogous to the well-known theorem on Artinian semigroup rings.

All Archimedean commutative semigroups S are described such that every S-homogeneous hereditary radical is S-normal. It is shown that this result is in a sense unimprovable.

A description of regular group rings is well known (see [12]). Various authors have considered regular semigroup rings (see [17], [8], [10], [11], [4]). These rings have been characterized for many important classes of semigroups, although the general problem turns out to be rather difficult and still has not got a complete solution. It seems natural to describe the regular radical in semigroup rings for semigroups of the classes mentioned. In [10], the regular semigroup rings of commutative semigroups were described. The aim of the present paper is to characterize the regular radical ρ(R[S]) for each associative ring R and commutative semigroup S.

Bands of associative rings were introduced in 1973 by Weissglass. For the radicals playing most essential roles in the structure theory (in particular, for those of Jacobson, Baer, Levitsky, Koethe) it is shown how to find the radical of a band of rings. The technique of the general Kurosh-Amitsur radical theory is used to consider many radicals simultaneously.

A cancellative commutative semigroup s and a hereditary radical ρ are constructed such that ρ is S-homogeneous but not S-normal. This answers a question which arose in the literature.