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A note on finitely generated ideal-simple commutative semirings

Vítězslav Kala, Tomáš Kepka (2008)

Commentationes Mathematicae Universitatis Carolinae

Many infinite finitely generated ideal-simple commutative semirings are additively idempotent. It is not clear whether this is true in general. However, to solve the problem, one can restrict oneself only to parasemifields.

Algèbres de polynômes tropicaux

Dominique Castella (2013)

Annales mathématiques Blaise Pascal

Nous continuons dans ce second article, l’étude des outils algébrique de l’algèbre de la caractéristique 1 : nous examinons plus spécialement ici les algèbres de polynômes sur un semi-corps idempotent. Ce travail est motivé par le développement de la géométrie tropicale qui apparaît comme étant la géométrie algébrique de l’algèbre tropicale. En fait l’objet algébrique le plus intéressant est l’image de l’algèbre de polynôme dans son semi-corps des fractions. Nous pouvons ainsi retrouver sur les...

An ideal-based zero-divisor graph of direct products of commutative rings

S. Ebrahimi Atani, M. Shajari Kohan, Z. Ebrahimi Sarvandi (2014)

Discussiones Mathematicae - General Algebra and Applications

In this paper, specifically, we look at the preservation of the diameter and girth of the zero-divisor graph with respect to an ideal of a commutative ring when extending to a finite direct product of commutative rings.

Clifford semifields

Mridul K. Sen, Sunil K. Maity, Kar-Ping Shum (2004)

Discussiones Mathematicae - General Algebra and Applications

It is well known that a semigroup S is a Clifford semigroup if and only if S is a strong semilattice of groups. We have recently extended this important result from semigroups to semirings by showing that a semiring S is a Clifford semiring if and only if S is a strong distributive lattice of skew-rings. In this paper, we introduce the notions of Clifford semidomain and Clifford semifield. Some structure theorems for these semirings are obtained.

Construction, properties and applications of finite neofields

Anthony Donald Keedwell (2000)

Commentationes Mathematicae Universitatis Carolinae

We give a short account of the construction and properties of left neofields. Most useful in practice seem to be neofields based on the cyclic group and particularly those having an additional divisibility property, called D-neofields. We shall give examples of applications to the construction of orthogonal latin squares, to the design of tournaments balanced for residual effects and to cryptography.

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