scholarly journals Pseudo-distributive near-rings

1975 ◽  
Vol 12 (3) ◽  
pp. 449-456 ◽  
Author(s):  
Henry E. Heatherly ◽  
Steve Ligh
Keyword(s):  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Matrix Rings ◽  
Basic Properties ◽  
Necessary And Sufficient

In the study of the theory of rings, matrix rings, group rings, algebras, and so on, play a very important role. However, the analogous systems may not exist in the theory of near-rings. Recently Ligh obtained a necessary and sufficient condition for the set of n × n matrices with entries from a near-ring to be a near-ring. This opens the door for the study of other structures such as group near-rings, algebras, and so on. In this paper we initiate a study of the basic properties of pseudo-distributive near-rings, which is exactly the class of near-rings needed to carry out the construction of matrix near-rings, group near-rings, polynomials with near-ring coefficients, and so on.

scholarly journals On separable extensions of group rings and quaternion rings

1978 ◽  
Vol 1 (4) ◽  
pp. 433-438
Author(s):  
George Szeto
Keyword(s):  
Commutative Ring ◽  
Full Description ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Ring Extension ◽  
Quaternion Algebras ◽  
Separable Group ◽  
Necessary And Sufficient ◽  
Separable Extensions

The purposes of the present paper are (1) to give a necessary and sufficient condition for the uniqueness of the separable idempotent for a separable group ring extensionRG(Rmay be a non-commutative ring), and (2) to give a full description of the set of separable idempotents for a quaternion ring extensionRQover a ringR, whereQare the usual quaternionsi,j,kand multiplication and addition are defined as quaternion algebras over a field. We shall show thatRGhas a unique separable idempotent if and only ifGis abelian, that there are more than one separable idempotents for a separable quaternion ringRQ, and thatRQis separable if and only if2is invertible inR.

scholarly journals Commutative feebly nil-clean group rings

2019 ◽  
Vol 11 (2) ◽  
pp. 264-270
Author(s):  
Peter V. Danchev
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Clean Rings ◽  
Unital Ring ◽  
Necessary And Sufficient

Abstract An arbitrary unital ring R is called feebly nil-clean if any its element is of the form q + e − f, where q is a nilpotent and e, f are idempotents with ef = fe. For any commutative ring R and any abelian group G, we find a necessary and sufficient condition when the group ring R(G) is feebly nil-clean only in terms of R, G and their sections. Our result refines establishments due to McGovern et al. in J. Algebra Appl. (2015) on nil-clean rings and Danchev-McGovern in J. Algebra (2015) on weakly nil-clean rings, respectively.

Comaximal factorization of lifting ideals

2020 ◽  
pp. 2250044
Author(s):  
Esmaeil Rostami ◽  
Sina Hedayat ◽  
Reza Nekooei ◽  
Somayeh Karimzadeh
Keyword(s):  
Commutative Ring ◽  
Proper Ideal ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
New Approach ◽  
Ideal Formula ◽  
Basic Properties ◽  
Necessary And Sufficient

A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is called lifting whenever idempotents of [Formula: see text] lift to idempotents of [Formula: see text]. In this paper, many of the basic properties of lifting ideals are studied and we prove and extend some well-known results concerning lifting ideals and lifting idempotents by a new approach. Furthermore, we give a necessary and sufficient condition for every proper ideal of a commutative ring to be a product of pairwise comaximal lifting ideals.

Properties of bisect-diagonal quadrilaterals

2017 ◽  
Vol 101 (551) ◽  
pp. 214-226 ◽  
Author(s):  
Martin Josefsson
Keyword(s):  
General Class ◽  
The Other ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Basic Properties ◽  
Necessary And Sufficient

The general class of quadrilaterals where one diagonal is bisected by the other diagonal has appeared very rarely in the geometrical literature, but they have been named several times in connection with quadrilateral classifications. Günter Graumann strangely gave these objects two different names in [1, pp. 192, 194]: sloping-kite and sliding-kite. A. Ramachandran called them slant kites in [2, p. 54] and Michael de Villiers called them bisecting quadrilaterals in [3, pp. 19, 206]. The latter is a pretty good name, although a bit confusing: what exactly is bisected?We have found no papers and only two books where any theorems on such quadrilaterals are studied. In each of the books, one necessary and sufficient condition for such quadrilaterals is proved (see Theorem 1 and 2 in the next section). The purpose of this paper is to investigate basic properties ofconvexbisecting quadrilaterals, but we have chosen to give them a slightly different name. Let us first remind the reader that a quadrilateral whose diagonals have equal lengths is called an equidiagonal quadrilateral and one whose diagonals are perpendicular is called an orthodiagonal quadrilateral.

scholarly journals Skew group rings which are Galois

2000 ◽  
Vol 23 (4) ◽  
pp. 279-283
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Galois Extension ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inner Automorphism ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Skew Group Ring

LetS*Gbe a skew group ring of a finite groupGover a ringS. It is shown that ifS*Gis anG′-Galois extension of(S*G)G′, whereG′is the inner automorphism group ofS*Ginduced by the elements inG, thenSis aG-Galois extension ofSG. A necessary and sufficient condition is also given for the commutator subring of(S*G)G′inS*Gto be a Galois extension, where(S*G)G′is the subring of the elements fixed under each element inG′.

Hecke **-algebras on locally compact hypergroups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Seyyed Mohammad Tabatabaie ◽  
Bentolhoda Sadathoseyni
Keyword(s):  
Compact Group ◽  
Hecke Algebras ◽  
Locally Compact Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Basic Properties ◽  
Necessary And Sufficient

AbstractIn this paper, for a discrete hypergroup K and its subhypergroup H, we initiate the related Hecke {*}-algebra which is an extension of the classical one, and study its basic properties. Especially, we give a necessary and sufficient condition (named (β)) for this algebra to be associative. Also, we show that this new structure is an associative {*}-algebra if and only if K is a locally compact group.

scholarly journals Support Neighbourly Edge Irregular Graphs

2019 ◽  
Vol 8 (3) ◽  
pp. 5329-5332
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Basic Properties ◽  
New Family ◽  
Necessary And Sufficient ◽  
Irregular Graphs

In this paper, we introduce a new family of irregular graphs called support neighbourly edge irregular graphs based on degree in edge sense. In any graph, the support of an edge is the sum of the edge degrees of its neighbours. A graph is said to be support neighbourly edge irregular(or simply SNEI), if any two adjacent edges have different support. Basic properties of theses graphs are studied. A necessary and sufficient condition for a graph to be SNEI has been obtained and several methods to construct SNEI graph from other graphs have been discussed.

scholarly journals Optimization of the reducible objective functions with monotone factors subject to FRI constraints defined with continuous t-norms

2018 ◽  
pp. 1-19
Keyword(s):  
Objective Function ◽  
Special Kind ◽  
Sufficient Condition ◽  
Objective Functions ◽  
Necessary And Sufficient Condition ◽  
Basic Properties ◽  
Binary Operator ◽  
Necessary And Sufficient ◽  
Decreasing Functions

In this paper, we investigate a special kind of optimization with fuzzy relational inequalities constraints where a continuous t-norm is considered as the fuzzy composition and the objective function can be expressed as in which and are increasing and decreasing functions, respectively, and is a commutative and monotone binary operator. Some basic properties have been extended a necessary and sufficient condition is presented to realize the feasibility of the problem. Also, an algorithm is given to optimize the objective function on the region of the FRI constraints. Finally, five examples are appended with two continuous t-norms, Lukasiewicz and Yager, and different objective functions, for illustrating.

scholarly journals GROUP RINGS OF COUNTABLE NON-ABELIAN LOCALLY FREE GROUPS ARE PRIMITIVE

2011 ◽  
Vol 21 (03) ◽  
pp. 409-431 ◽  
Author(s):  
TSUNEKAZU NISHINAKA
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Free Groups ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hnn Extensions ◽  
Necessary And Sufficient ◽  
Countably Infinite

We prove that every group ring of a non-abelian locally free group which is the union of an ascending sequence of free groups is primitive. In particular, every group ring of a countable non-abelian locally free group is primitive. In addition, by making use of the result, we give a necessary and sufficient condition for group rings of ascending HNN extensions of free groups to be primitive, which extends the main result in [Group rings of proper ascending HNN extensions of countably infinite free groups are primitive, J. Algebra317 (2007) 581–592] to the general cardinality case.

(G) Fuzzy Integral on Fuzzy Set

2014 ◽  
Vol 926-930 ◽  
pp. 2863-2866
Author(s):  
Gui Ling Li
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Convergence Theorem ◽  
Fuzzy Integral ◽  
Monotone Convergence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Basic Properties ◽  
Monotone Convergence Theorem ◽  
Necessary And Sufficient

In this paper, the concept of (G) fuzzy integral on a fuzzy set is given and the basic properties of it are discussed. Monotone Convergence Theorem and Fatou Lemma are proved. Finally, A necessary and sufficient condition on which (G) fuzzy integrals of two fuzzy measurable functions on fuzzy sets are always equal is given.

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