Invertible matrices over a class of semirings

2021 ◽  
Author(s):  
David Dolžan
Keyword(s):  
Nilpotent Elements ◽  
Commuting Graph ◽  
Invertible Elements ◽  
Invertible Matrices

We characterize the invertible matrices over a class of semirings such that the set of additively invertible elements is equal to the set of nilpotent elements. We achieve this by studying the liftings of the orthogonal sums of elements that are “almost idempotent” to those that are idempotent. Finally, we show an application of the obtained results to calculate the diameter of the commuting graph of the group of invertible matrices over the semirings in question.

scholarly journals Non-stable $K$-theory for $QB$-rings

2007 ◽  
Vol 100 (2) ◽  
pp. 265
Author(s):  
Pere Ara ◽  
Francesc Perera
Keyword(s):  
Large Class ◽  
Projective Modules ◽  
Finitely Generated ◽  
Cancellation Condition ◽  
K Theory ◽  
Invertible Elements ◽  
Invertible Matrices

We study the class of $QB$-rings that satisfy the weak cancellation condition of separativity for finitely generated projective modules. This property turns out to be crucial for proving that all (quasi-)invertible matrices over a $QB$-ring can be diagonalised using row and column operations. The main two consequences of this fact are: (i) The natural map $(\mathrm{GL}_1(R)\to K_1(R)$ is surjective, and (ii) the only obstruction to lift invertible elements from a quotient is of $K$-theoretical nature. We also show that for a reasonably large class of $QB$-rings that includes the prime ones, separativity always holds.

On the Topologically Invertible Elements of a Topological Algebra

2007 ◽  
Vol 107 (1) ◽  
pp. 73-80
Author(s):  
Hugo Arizmendi-Peimbert ◽  
Angel Carrillo-Hoyo
Keyword(s):  
Topological Algebra ◽  
Invertible Elements

On quasi-radical of near-ring of polynomials

2019 ◽  
Vol 56 (2) ◽  
pp. 252-259
Author(s):  
Ebrahim Hashemi ◽  
Fatemeh Shokuhifar ◽  
Abdollah Alhevaz
Keyword(s):  
Commutative Ring ◽  
Nilpotent Elements ◽  
Ring Of Polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

EXTENDING RESULTS OF MORGAN AND PARKER ABOUT COMMUTING GRAPHS

2021 ◽  
pp. 1-9
Author(s):  
NICOLAS F. BEIKE ◽  
RACHEL CARLETON ◽  
DAVID G. COSTANZO ◽  
COLIN HEATH ◽  
MARK L. LEWIS ◽  
...  
Keyword(s):  
Frobenius Group ◽  
Connected Components ◽  
Commuting Graph

Abstract Morgan and Parker proved that if G is a group with ${\textbf{Z}(G)} = 1$ , then the connected components of the commuting graph of G have diameter at most $10$ . Parker proved that if, in addition, G is solvable, then the commuting graph of G is disconnected if and only if G is a Frobenius group or a $2$ -Frobenius group, and if the commuting graph of G is connected, then its diameter is at most $8$ . We prove that the hypothesis $Z (G) = 1$ in these results can be replaced with $G' \cap {\textbf{Z}(G)} = 1$ . We also prove that if G is solvable and $G/{\textbf{Z}(G)}$ is either a Frobenius group or a $2$ -Frobenius group, then the commuting graph of G is disconnected.

scholarly journals Subgroups of finite index in an additive group of a ring

2001 ◽  
Vol 27 (2) ◽  
pp. 83-89
Author(s):  
Doostali Mojdeh ◽  
S. Hassan Hashemi
Keyword(s):  
Finite Index ◽  
Additive Group ◽  
Infinite Field ◽  
Division Rings ◽  
Invertible Elements

IfKis an infinite field andG⫅Kis a subgroup of finite index in an additive group, thenK∗=G∗G∗−1whereG∗denotes the set of all invertible elements inGandG∗−1denotes all inverses of elements ofG∗. Similar results hold for various fields, division rings and rings.

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

scholarly journals Commutativity and structure of rings with commuting nilpotents

1983 ◽  
Vol 6 (1) ◽  
pp. 119-124
Author(s):  
Hazar Abu-Khuzam ◽  
Adil Yaqub
Keyword(s):  
Commutative Rings ◽  
Structure Theory ◽  
Nilpotent Elements

LetRbe a ring and letNdenote the set of nilpotent elements ofR. LetZdenote the center ofR. Suppose that (i)Nis commutative, (ii) for everyxinRthere existsx′ϵ<x>such thatx−x2x′ϵN, where<x>denotes the subring generated byx, (iii) for everyx,yinR, there exists an integern=n(x,y)≥1such that both(xy)n−(yx)nand(xy)n+1−(yx)n+1belong toZ. ThenRis commutative and, in fact,Ris isomorphic to a subdirect sum of nil commutative rings and local commutative rings. It is further shown that both conditions in hypothesis (iii) are essential. The proof uses the structure theory of rings along with some earlier results of the authors.

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