On quasi-prime differential semiring ideals

• Main Author: І. О. Мельник
• Format: Article
• Language: English
• Published: State University “Uzhhorod National University” 2020-11-01
• Series: Науковий вісник Ужгородського університету. Серія: Математика і інформатика
• Subjects: differential semiring; differential ideal; semiring ideal; quasi-prime ideal
• Online Access: http://visnyk-math.uzhnu.edu.ua/article/view/214483

The notion of a \verb"quasi-prime ideal", for the first time, was introduced in differential commutative rings, i.e. commutative rings considered together with a derivation, as differential ideals maximal among those not meeting some multiplicatively closed subset of a ring. The notion of a semiring derivation is traditionally defined as an additive map satisfying the Leibnitz rule. Due to rapid development of semiring theory in recent years, the need of considering ideals in semirings defined by similar conditions arose. The present paper is devoted to investigating the notion of a \verb"quasi-prime ideal" of differential semiring (which is defined as a semiring together with a derivation on it), not necessarily commutative. It aims to show, how \verb"quasi-prime ideals" are related to prime differential ideals, primary ideals, maximal ideals and other types of ideals of semirings. The paper consists of two main parts. In the first part, the author investigates some properties of quasi-prime differential ideals, and gives some examples of such semiring ideals, such as prime differential, maximal differential ideals, or ideal obtained by derivation operator acting on a prime ideal of a semiring. It contains a theorem, which gives equivalent conditions for a quasi-prime semiring ideal to be prime. The second part of the paper is devoted to considering chains of \verb"quasi-prime ideals". In this part, the interrelation between \verb"quasi-prime ideals" and other types of differential ideals of semirings is established. It contains a theorem, which gives a characterization of such ideals in case of a commutative semiring. This characterization involves the notion of the radical of an ideal of a semiring and a derivation operator for semirings. The paper ends with a theorem, which states that every chain of quasi-prime ideals of a semiring has the least upper bound and the greatest lower bound. It is also proven that every \verb"quasi-prime ideal" containing some differential ideal contains a \verb"quasi-prime ideal" minimal among all the quasi-prime ideals of the given semiring, which contain the above mentioned differential ideal.