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Journal ArticleDOI

On the Jacobson radical of a semiring

Kenzo Iizuka1
01 Jan 1959-Tohoku Mathematical Journal (Mathematical Institute, Tohoku University)-Vol. 11, Iss: 3, pp 409-421
About: This article is published in Tohoku Mathematical Journal.The article was published on 1959-01-01 and is currently open access. It has received 91 citations till now. The article focuses on the topics: Jacobson radical & Semiring.

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Journal ArticleDOI
TL;DR: This paper initiates the study of soft semirings by using the soft set theory, and the notions of soft Semirings, soft subsemirings,soft ideals, idealistic softSemirings and soft semiring homomorphisms are introduced, and several related properties are investigated.
Abstract: Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we initiate the study of soft semirings by using the soft set theory. The notions of soft semirings, soft subsemirings, soft ideals, idealistic soft semirings and soft semiring homomorphisms are introduced, and several related properties are investigated.

579 citations


Cites background from "On the Jacobson radical of a semiri..."

  • ...Consequently, some more restricted concepts of ideals such as k-ideals [16] and h-ideals [17] have been introduced in the study of the semiring theory....

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Journal ArticleDOI
01 May 2017
TL;DR: The concept of soft rough hemirings is introduced, which is an extended notion of a rough hemiring, which provides a new research idea for soft rough algebraic research.
Abstract: Graphical abstractDisplay Omitted In this paper, we investigate the relationships among rough sets, soft sets and hemirings. The concept of soft rough hemirings is introduced, which is an extended notion of a rough hemiring. It is pointed out that in this paper, we first apply soft rough sets to algebraic structure-hemirings. Further, we first put forward the concepts of C-soft sets and CC-soft sets, which provide a new research idea for soft rough algebraic research. Moreover, we study roughness in hemirings with respect to MSR-approximation spaces. Some new soft rough operations over hemirings are explored. In particular, lower and upper MSR-hemirings (k-ideal and h-ideal) are investigated. Finally, we put forth an approach for multicriteria group decision making problem based on modified soft rough sets and offer an actual example.

158 citations

Journal ArticleDOI
TL;DR: The fuzzy setting of a left h-ideal in a hemiring is constructed, and basic properties are investigated.

111 citations


Cites background from "On the Jacobson radical of a semiri..."

  • ...However, a definition of ideal in any additively commutative semiring S can be given which coincides with Iizuka’s definition provided S is a hemiring, and it is called h-ideal....

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  • ...A still more restricted class of ideals in hemirings has been given by Iizuka [12]....

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Journal ArticleDOI
TL;DR: The aim of this paper is to introduce the notion of a rough soft hemiring, which is an extended notion ofa rough hemiring and a soft hemires, and to study roughness in soft hemirings with respect to Pawlak approximation spaces.
Abstract: The aim of this paper is to introduce the notion of a rough soft hemiring, which is an extended notion of a rough hemiring and a soft hemiring. We study roughness in soft hemirings with respect to Pawlak approximation spaces. Some new rough soft operations are explored. In particular, lower and upper rough soft hemirings and k-idealistic, h-idealistic, strong h-idealistic idealistic soft hemirings are investigated. Finally, an important result on an upper strong h-idealistic rough soft hemiring via Bourne's congruence relations are obtained.

110 citations

Journal ArticleDOI
TL;DR: It is proved that a fuzzy subset @z of a hemiring S is a prime fuzzy left (right) h-ideal of S if and only if @z is two-valued, @z(0)=1, and the set of all x in S such that @z

103 citations


Cites background from "On the Jacobson radical of a semiri..."

  • ...Another more restricted class of ideals has been given in hemirings by Iizuka [11]....

    [...]