Morphic Properties of Extensions of Rings

2010 ◽  
Vol 17 (02) ◽  
pp. 337-344
Author(s):  
Qinghe Huang ◽  
Jianlong Chen
Keyword(s):  
Positive Integer ◽  
Cyclic Group ◽  
Group Ring ◽  
Regular Ring ◽  
Strongly Regular ◽  
Left Annihilator

An element a in a ring R is called left morphic if R/Ra ≅ ℓR(a), where ℓR(a) denotes the left annihilator of a in R. A ring R is said to be left morphic if every element is left morphic. In this paper, it is shown that if I is an ideal of a unit regular ring R, then for each positive integer n, [Formula: see text] is a left morphic ring. This extends two recent results of Lee and Zhou. It is also proved that if R is a strongly regular ring and Cn= 〈g〉 is a cyclic group of order n ≥ 2, then for any r ∈ R, 1 + rg is morphic in the group ring RCn.

scholarly journals STRONGLY CLEAN POWER SERIES RINGS

2007 ◽  
Vol 50 (1) ◽  
pp. 73-85 ◽  
Author(s):  
Jianlong Chen ◽  
Yiqiang Zhou
Keyword(s):  
Finite Group ◽  
Power Series ◽  
Group Ring ◽  
Regular Ring ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Regular Elements ◽  
Strongly Regular ◽  
Power Series Rings

AbstractAn element $a$ in a ring $R$ with identity is called strongly clean if it is the sum of an idempotent and a unit that commute. And $a\in R$ is called strongly $\pi$-regular if both chains $aR\supseteq a^2R\supseteq\cdots$ and $Ra\supseteq Ra^2\supseteq\cdots$ terminate. A ring $R$ is called strongly clean (respectively, strongly $\pi$-regular) if every element of $R$ is strongly clean (respectively, strongly $\pi$-regular). Strongly $\pi$-regular elements of a ring are all strongly clean. Let $\sigma$ be an endomorphism of $R$. It is proved that for $\varSigma r_ix^i\in R[[x,\sigma]]$, if $r_0$ or $1-r_0$ is strongly $\pi$-regular in $R$, then $\varSigma r_ix^i$ is strongly clean in $R[[x,\sigma]]$. In particular, if $R$ is strongly $\pi$-regular, then $R[[x,\sigma]]$ is strongly clean. It is also proved that if $R$ is a strongly $\pi$-regular ring, then $R[x,\sigma]/(x^n)$ is strongly clean for all $n\ge1$ and that the group ring of a locally finite group over a strongly regular or commutative strongly $\pi$-regular ring is strongly clean.

scholarly journals Commutative semigroups whose endomorphisms are power functions

2021 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Positive Integer ◽  
Power Function ◽  
Cyclic Group ◽  
Commutative Semigroup ◽  
Commutative Monoid ◽  
Power Functions ◽  
Commutative Semigroups ◽  
Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

scholarly journals Extension of Strongly Regular Graphs

10.37236/878 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ralucca Gera ◽  
Jian Shen
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Regularity Property ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Strongly Regular ◽  
Vertex Degrees

The Friendship Theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. In this paper, we generalize the Friendship Theorem. Let $\lambda$ be any nonnegative integer and $\mu$ be any positive integer. Suppose each pair of friends have exactly $\lambda$ common friends and each pair of strangers have exactly $\mu$ common friends in a party. The corresponding graph is a generalization of strongly regular graphs obtained by relaxing the regularity property on vertex degrees. We prove that either everyone has exactly the same number of friends or there exists a politician who is a friend of everybody. As an immediate consequence, this implies a recent conjecture by Limaye et. al.

On a Sheaf of Division Rings*

1975 ◽  
Vol 17 (5) ◽  
pp. 727-731
Author(s):  
George Szeto
Keyword(s):  
Principal Ideal ◽  
Regular Ring ◽  
Central Idempotent ◽  
Division Rings ◽  
Maximal Ideals ◽  
Strongly Regular ◽  
Th 1

R. Arens and I. Kaplansky ([1]) call a ring A biregular if every two sided principal ideal of A is generated by a central idempotent and a ring A strongly regular if for any a in A there exists b in A such that a=a2b. In ([1], Sections 2 and 3), a lot of interesting properties of a biregular ring and a strongly regular ring are given. Some more properties can also be found in [3], [5], [8], [9] and [13]. For example, J. Dauns and K. Hofmann ([3]) show that a biregular ring A is isomorphic with the global sections of the sheaf of simple rings A/K where K are maximal ideals of A. The converse is also proved by R. Pierce ([9], Th. 11–1). Moreover, J. Lambek ([5], Th. 1) extends the above representation of a biregular ring to a symmetric module.

On semireversible rings

2019 ◽  
pp. 2150018
Author(s):  
Debraj Roy ◽  
Tikaram Subedi
Keyword(s):  
Positive Integer ◽  
Matrix Rings ◽  
New Class ◽  
Ring Of Fractions ◽  
Strongly Regular

In this paper, we introduce and study a new class of rings which we call semireversible rings. A ring [Formula: see text] is called semireversible if for any [Formula: see text] implies there exists a positive integer [Formula: see text] such that [Formula: see text]. We observe that the class of semireversible rings strictly lies between the class of central reversible rings and weakly reversible rings. Some relations are provided between semireversible rings and many other known classes of rings. Some extensions of semireversible rings such as ring of fractions, Dorroh extension, subrings of matrix rings are investigated. Finally, we study semireversible rings via modules over them wherein among other results, we prove that a semireversible left (right) SF-ring is strongly regular.

scholarly journals Sums of element orders in groups of odd order

2021 ◽  
pp. 1-15
Author(s):  
Marcel Herzog ◽  
Patrizia Longobardi ◽  
Mercede Maj
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Cyclic Group ◽  
Element Orders ◽  
Maximum Value ◽  
Odd Order ◽  
A Finite Group

For a finite group [Formula: see text], let [Formula: see text] denote the sum of element orders of [Formula: see text]. If [Formula: see text] is a positive integer let [Formula: see text] be the cyclic group of order [Formula: see text]. It is known that [Formula: see text] is the maximum value of [Formula: see text] on the set of groups of order [Formula: see text], and [Formula: see text] if and only if [Formula: see text] is cyclic of order [Formula: see text]. In this paper, we investigate the second largest value of [Formula: see text] on the set of groups of order [Formula: see text] and the structure of groups [Formula: see text] of order [Formula: see text] with this value of [Formula: see text] when [Formula: see text] is odd.

scholarly journals Monochromatic and Zero-Sum Sets of Nondecreasing Modified Diameter

10.37236/1054 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
David Grynkiewicz ◽  
Rasheed Sabar
Keyword(s):  
Positive Integer ◽  
Cyclic Group ◽  
Prime Divisor ◽  
Zero Sum ◽  
Equality Holding

Let $m$ be a positive integer whose smallest prime divisor is denoted by $p$, and let ${\Bbb Z}_m$ denote the cyclic group of residues modulo $m$. For a set $B=\{x_1,x_2,\ldots,x_m\}$ of $m$ integers satisfying $x_1 < x_2 < \cdots < x_m$, and an integer $j$ satisfying $2\leq j \leq m$, define $g_j(B)=x_j-x_1$. Furthermore, define $f_j(m,2)$ (define $f_j(m,{\Bbb Z}_m)$) to be the least integer $N$ such that for every coloring $\Delta : \{1,\ldots,N\}\rightarrow \{0,1\}$ (every coloring $\Delta : \{1,\ldots,N\}\rightarrow {\Bbb Z}_m$), there exist two $m$-sets $B_1,B_2\subset \{1,\ldots,N\}$ satisfying: (i) $\max(B_1) < \min(B_2)$, (ii) $g_j(B_1)\leq g_j(B_2)$, and (iii) $|\Delta (B_i)|=1$ for $i=1,2$ (and (iii) $\sum_{x\in B_i}\Delta (x)=0$ for $i=1,2$). We prove that $f_j(m,2)\leq 5m-3$ for all $j$, with equality holding for $j=m$, and that $f_j(m,{\Bbb Z}_m)\leq 8m+{m\over p}-6$. Moreover, we show that $f_j(m,2)\ge 4m-2+(j-1)k$, where $k=\left\lfloor\left(-1+\sqrt{{8m-9+j\over j-1}}\right){/2}\right\rfloor$, and, if $m$ is prime or $j\geq{m\over p}+p-1$, that $f_j(m,{\Bbb Z}_m)\leq 6m-4$. We conclude by showing $f_{m-1}(m,2)=f_{m-1}(m,{\Bbb Z}_m)$ for $m\geq 9$.

Lehmer’s problem for arbitrary groups

2020 ◽  
pp. 1-32
Author(s):  
W. Lück
Keyword(s):  
Cyclic Group ◽  
Group Ring ◽  
Bounded Operator ◽  
Integral Group Ring ◽  
Integral Group ◽  
Infinite Cyclic Group ◽  
Matrix Formula ◽  
Lehmer’S Problem

We consider the problem whether for a group [Formula: see text] there exists a constant [Formula: see text] such that for any [Formula: see text]-matrix [Formula: see text] over the integral group ring [Formula: see text] the Fuglede–Kadison determinant of the [Formula: see text]-equivariant bounded operator [Formula: see text] given by right multiplication with [Formula: see text] is either one or greater or equal to [Formula: see text]. If [Formula: see text] is the infinite cyclic group and we consider only [Formula: see text], this is precisely Lehmer’s problem.

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