On quasi-commutative rings

정다운,김병옥,김홍기,이양,남상복,류성주,성효진,윤상조 대한수학회 2016 대한수학회지 Vol.53 No.2

We study the structure of central elements in relation with polynomial rings and introduce {\it quasi-commutative}as a generalization of commutative rings. The Jacobson radical of the polynomial ring over a quasi-commutative ring is shown to coincide with the set of all nilpotent polynomials; and locally finite quasi-commutative rings are shown to be commutative. We also provide several sorts of examples by showing the relations between quasi-commutative rings and other ring properties which have roles in ring theory. We examine next various sorts of ring extensions of quasi-commutative rings.

ON QUASI-COMMUTATIVE RINGS

Jung, Da Woon,Kim, Byung-Ok,Kim, Hong Kee,Lee, Yang,Nam, Sang Bok,Ryu, Sung Ju,Sung, Hyo Jin,Yun, Sang Jo Korean Mathematical Society 2016 대한수학회지 Vol.53 No.2

We study the structure of central elements in relation with polynomial rings and introduce quasi-commutative as a generalization of commutative rings. The Jacobson radical of the polynomial ring over a quasi-commutative ring is shown to coincide with the set of all nilpotent polynomials; and locally finite quasi-commutative rings are shown to be commutative. We also provide several sorts of examples by showing the relations between quasi-commutative rings and other ring properties which have roles in ring theory. We examine next various sorts of ring extensions of quasi-commutative rings.

COMMUTATIVITY OF ASSOCIATION SCHEMES OF ORDER pq

Akihide Hanaki,히라사카 영남수학회 2013 East Asian mathematical journal Vol.29 No.1

Let (X; S) be an association scheme where X is a nite set and S is a partition of X X. The size of X is called the order of (X; S). We de ne C to be the set of positive integers m such that each association scheme of order m is commutative. It is known that each prime is belonged to C and it is conjectured that each prime square is belonged to C. In this article we give a su cient condition for a scheme of order pq to be commutative where p and q are primes, and obtain a partial answer for the conjecture in case where p = q.

COMMUTATIVITY OF ASSOCIATION SCHEMES OF ORDER pq

Hanaki, Akihide,Hirasaka, Mitsugu The Youngnam Mathematical Society 2013 East Asian mathematical journal Vol.29 No.1

Let (X, S) be an association scheme where X is a finite set and S is a partition of $X{\times}X$. The size of X is called the order of (X, S). We define $\mathcal{C}$ to be the set of positive integers m such that each association scheme of order $m$ is commutative. It is known that each prime is belonged to $\mathcal{C}$ and it is conjectured that each prime square is belonged to $\mathcal{C}$. In this article we give a sufficient condition for a scheme of order pq to be commutative where $p$ and $q$ are primes, and obtain a partial answer for the conjecture in case where $p=q$.

RING WHOSE MAXIMAL ONE-SIDED IDEALS ARE TWO-SIDED

Huh, Chan,Jang, Sung-Hee,Kim, Chol-On,Lee, Yang Korean Mathematical Society 2002 대한수학회보 Vol.39 No.3

In this note we are concerned with relationships between one-sided ideals and two-sided ideals, and study the properties of polynomial rings whose maximal one-sided ideals are two-sided, in the viewpoint of the Nullstellensatz on noncommutative rings. Let R be a ring and R[x] be the polynomial ring over R with x the indeterminate. We show that eRe is right quasi-duo for $0{\neq}e^2=e{\in}R$ if R is right quasi-duo; R/J(R) is commutative with J(R) the Jacobson radical of R if R[$\chi$] is right quasi-duo, from which we may characterize polynomial rings whose maximal one-sided ideals are two-sided; if R[x] is right quasi-duo then the Jacobson radical of R[x] is N(R)[x] and so the $K\ddot{o}the's$ conjecture (i.e., the upper nilradical contains every nil left ideal) holds, where N(R) is the set of all nilpotent elements in R. Next we prove that if the polynomial rins R[x], over a reduced ring R with $\mid$X$\mid$ $\geq$ 2, is right quasi-duo, then R is commutative. Several counterexamples are included for the situations that occur naturally in the process of this note.

Quasi-commutativity related to powers

김현민,Dan Li,Zhelin Piao 대한수학회 2017 대한수학회보 Vol.54 No.6

We study the quasi-commutativity in relation with powers of coefficients of polynomials. In the procedure we introduce the concept of {\it $\pi$-quasi-commutative} ring as a generalization of quasi-commutative rings. We show first that every $\pi$-quasi-commutative ring is Abelian and that a locally finite Abelian ring is $\pi$-quasi-commutative. The role of these facts are essential to our study in this note. The structures of various sorts of $\pi$-quasi-commutative rings are investigated to answer the questions raised naturally in the process, in relation to the structure of Jacobson and nil radicals.

INTUITIONISTIC FUZZY COMMUTATIVE IDEALSOF BCK-ALGEBRAS

전영배,이동수,박철환 한국수학교육학회 2008 純粹 및 應用數學 Vol.15 No.1

After the introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy sets introduced by Aranassov is one among them. In this paper, we apply the concept of an intuitionistic fuzzy set to commutative ideals in BCK-algebras. The notion of an intuitionistic fuzzy commutative ideal of a BCK-algebra is introduced, and some related properties are investigated. Characterizations of an intuitionistic fuzzy commutative ideal are given. Conditions for an intuitionistic fuzzy ideal to be an intuitionistic fuzzy commutative ideal are given. Using a collection of commutative ideals, intuitionistic fuzzy commutative ideals are established.

QUASI-COMMUTATIVITY RELATED TO POWERS

Kim, Hyun-Min,Li, Dan,Piao, Zhelin Korean Mathematical Society 2017 대한수학회보 Vol.54 No.6

We study the quasi-commutativity in relation with powers of coefficients of polynomials. In the procedure we introduce the concept of ${\pi}$-quasi-commutative ring as a generalization of quasi-commutative rings. We show first that every ${\pi}$-quasi-commutative ring is Abelian and that a locally finite Abelian ring is ${\pi}$-quasi-commutative. The role of these facts are essential to our study in this note. The structures of various sorts of ${\pi}$-quasi-commutative rings are investigated to answer the questions raised naturally in the process, in relation to the structure of Jacobson and nil radicals.

Dokdo commutative ideals of BCK-algebras

이정곤,Samy M. Mostafa,허걸,전영배 원광대학교 기초자연과학연구소 2022 ANNALS OF FUZZY MATHEMATICS AND INFORMATICS Vol.23 No.1

The purpose of this paper is to study by applying Dokdo structure to commutative ideal in BCK-algebras. The notion of Dokdo commutative ideal is introduced, and their properties are investigated. The relationship between Dokdo ideal and Dokdo commutative ideal is discussed. Example to show that a Dokdo ideal may not be a Dokdo com- mutative ideal is provided, and then the conditions under which a Dokdo ideal can be a Dokdo commutative ideal are explored. Conditions for a Dokdo structure to be a Dokdo commutative ideal are provided, and char- acterizations of a Dokdo commutative ideal are displayed. Finally, the extension property for a Dokdo commutative ideal is established.

Falling Fuzzy BCI-Commutative Ideals

( Young Bae Jun ),( Seok Zun Song ) 호남수학회 2014 호남수학학술지 Vol.36 No.3

On the basis of the theory of a falling shadow and fuzzy sets, the notion of a falling fuzzy BCI-commutative ideal of a BCI-algebra is introduced. Relations between falling fuzzy BCI-commutative ideals and falling fuzzy ideals are given. Relations between fuzzy BCI-commutative ideals and falling fuzzy BCI-commutative ideals are provided. Characterizations of a falling fuzzy BCI-commutative ideal are established, and conditions for a falling fuzzy (closed) ideal to be a falling fuzzy BCI-commutative ideal are considered.