scholarly journals Characterizing Some Rings of Finite Order

2021 ◽  
Vol 53 ◽  
Author(s):  
Rajat Kanti Nath ◽  
Jutirekha Dutta ◽  
Dhiren Basnet
Keyword(s):  
Positive Integer ◽  
Finite Order ◽  
Finite Ring ◽  
Finite Rings

In this paper, we compute the number of distinct centralizers of some classes of finite rings. We then characterize all finite rings with $n$ distinct centralizers for any positive integer $n \le 5$. Further we give some connections between the number of distinct centralizers of a finite ring and its commutativity degree.

scholarly journals On orders of directly indecomposable finite rings

1992 ◽  
Vol 46 (3) ◽  
pp. 353-359 ◽  
Author(s):  
Yasuyuki Hirano ◽  
Takao Sumiyama
Keyword(s):  
Positive Integer ◽  
Finite Ring ◽  
Finite Rings ◽  
Simple Ring ◽  
Nilpotent Ring ◽  
Group Of Units

Let R be a directly indecomposable finite ring. Let p be a prime, let m be a positive integer and suppose the radical of R has pm elements. Then we show that . As a consequence, we have that, for a given finite nilpotent ring N, there are up to isomorphism only finitely many finite rings not having simple ring direct summands, with radical isomorphic to N. Let R* denote the group of units of R. Then we prove that (1 − 1/p)m+1 ≤ |R*| / |R| ≤ 1 − 1/pm. As a corollary, we obtain that if R is a directly indecomposable non-simple finite 2′-ring then |R| < |R*| |Rad(R)|.

scholarly journals Unit groups of cube radical zero commutative completely primary finite rings

2005 ◽  
Vol 2005 (4) ◽  
pp. 579-592
Author(s):  
Chiteng'a John Chikunji
Keyword(s):  
Positive Integer ◽  
Maximal Ideal ◽  
Finite Ring ◽  
Finitely Generated ◽  
Finite Rings ◽  
Group Of Units ◽  
Generating Sets ◽  
Zero Divisors ◽  
Unique Maximal Ideal

A completely primary finite ring is a ringRwith identity1≠0whose subset of all its zero-divisors forms the unique maximal idealJ. LetRbe a commutative completely primary finite ring with the unique maximal idealJsuch thatJ3=(0)andJ2≠(0). ThenR/J≅GF(pr)and the characteristic ofRispk, where1≤k≤3, for some primepand positive integerr. LetRo=GR(pkr,pk)be a Galois subring ofRand let the annihilator ofJbeJ2so thatR=Ro⊕U⊕V, whereUandVare finitely generatedRo-modules. Let nonnegative integerssandtbe numbers of elements in the generating sets forUandV, respectively. Whens=2,t=1, and the characteristic ofRisp; and whent=s(s+1)/2, for any fixeds, the structure of the group of unitsR∗of the ringRand its generators are determined; these depend on the structural matrices(aij)and on the parametersp,k,r, ands.

Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 126
Author(s):  
Hong Li ◽  
Hongyan Xu
Keyword(s):  
Positive Integer ◽  
Finite Order ◽  
Difference Equations ◽  
Functional Equations ◽  
Growth Order ◽  
Entire Solutions ◽  
Partial Differential ◽  
Integer Order ◽  
Significant Difference ◽  
Quadratic Trinomial

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z12=eg(z),f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z2+∂f(z)∂z1+∂f(z)∂z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang.

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

scholarly journals THE COMPLEXITY OF THE EQUIVALENCE PROBLEM OVER FINITE RINGS

2011 ◽  
Vol 54 (1) ◽  
pp. 193-199 ◽  
Author(s):  
GÁBOR HORVÁTH
Keyword(s):  
Polynomial Time ◽  
Jacobson Radical ◽  
Equivalence Problem ◽  
Finite Ring ◽  
Polynomial Equations ◽  
Finite Rings

AbstractWe investigate the complexity of the equivalence problem over a finite ring when the input polynomials are written as sum of monomials. We prove that for a finite ring if the factor by the Jacobson radical can be lifted in the centre, then this problem can be solved in polynomial time. This result provides a step in proving a dichotomy conjecture of Lawrence and Willard (J. Lawrence and R. Willard, The complexity of solving polynomial equations over finite rings (manuscript, 1997)).

On strongly right bounded finite rings

1991 ◽  
Vol 44 (3) ◽  
pp. 353-355 ◽  
Author(s):  
Weimin Xue
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Nonzero Ideal ◽  
Finite Ring ◽  
Finite Rings

An associative ring is called strongly right (left) bounded if every nonzero right (left) ideal contains a nonzero ideal. We prove that if R is a strongly right bounded finite ring with unity and the order |R| of R has no factors of the form p5, then R is strongly left bounded. This answers a question of Birkenmeier and Tucci.

scholarly journals Unit Groups of Classes of Five Radical Zero Commutative Completely Primary Finite Rings

2021 ◽  
pp. 137-154
Author(s):  
Hezron Saka Were ◽  
Maurice Oduor Owino ◽  
Moses Ndiritu Gichuki
Keyword(s):  
Maximal Ideal ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisors ◽  
Unique Maximal Ideal

In this paper, R is considered a completely primary finite ring and Z(R) is its subset of all zero divisors (including zero), forming a unique maximal ideal. We give a construction of R whose subset of zero divisors Z(R) satisfies the conditions (Z(R))5 = (0); (Z(R))4 ̸= (0) and determine the structures of the unit groups of R for all its characteristics.

Finite Rings in Which 1 is a Sum of Two Non-p-thPower Units

1976 ◽  
Vol 28 (1) ◽  
pp. 94-103 ◽  
Author(s):  
David Jacobson
Keyword(s):  
Prime Number ◽  
Finite Ring ◽  
Finite Rings ◽  
Group Of Units

LetRbe a finite ring with 1 and letR*denote the group of units ofR.Letpbe a prime number. In this paper we consider the question of whether there exista, binR*such thataandb arenon-p-th powers whose sum is 1. If such units a,bexisting, we say that R is an N (p)-ring. Of course ifpdoes not divide |R*|, the order of R*, then every element inR*is apthpower.

Linear Homogeneous Equations Over Finite Rings

1964 ◽  
Vol 16 ◽  
pp. 532-538 ◽  
Author(s):  
Harlan Stevens
Keyword(s):  
Finite Ring ◽  
Finite Rings ◽  
Image Position

The intent of this paper is to apply the following theorem in several particular instances:Theorem 1. For any finite ring of q elements, let {} be a collection of s subsets of , each containing hi (i = 1, 2, . . . , s) members, and let denote the set of all differences d′ — d″ with d′ and d″ from including d′ = d″. Furthermore, suppose that

scholarly journals On Generalized Non-commuting Graph of a Finite Ring

2018 ◽  
Vol 25 (01) ◽  
pp. 149-160 ◽  
Author(s):  
Jutirekha Dutta ◽  
Dhiren K. Basnet ◽  
Rajat K. Nath
Keyword(s):  
Simple Graph ◽  
Finite Ring ◽  
Dominating Sets ◽  
Finite Rings ◽  
Commuting Graph ◽  
Vertex Set

Let S and K be two subrings of a finite ring R. Then the generalized non-commuting graph of subrings S, K of R, denoted by ГS,K, is a simple graph whose vertex set is [Formula: see text], and where two distinct vertices a, b are adjacent if and only if [Formula: see text] or [Formula: see text] and [Formula: see text]. We determine the diameter, girth and some dominating sets for ГS,K. Some connections between ГS,K and Pr(S, K) are also obtained. Further, ℤ-isoclinism between two pairs of finite rings is defined, and we show that the generalized non-commuting graphs of two ℤ-isoclinic pairs are isomorphic under some conditions.

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