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.

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.

scholarly journals On the Zero Divisor Graphs of a Class of Commutative Completely Primary Finite Rings

2016 ◽  
Vol 12 (3) ◽  
pp. 6021-6026
Author(s):  
Maurice Oduor ◽  
Walwenda Shadrack Adero
Keyword(s):  
Maximal Ideal ◽  
Zero Divisor ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisors ◽  
Unique Maximal Ideal

Let R be a Completely Primary Finite Ring with a unique maximal ideal Z(R)), satisfying ((Z(R))n−1 ̸= (0) and (Z(R))n = (0): The structures of the units some classes of such rings have been determined. In this paper, we investigate the structures of the zero divisors of R:

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 Finite completely primary rings in which the product of any two zero divisors of a ring is in its coefficient subring

1994 ◽  
Vol 17 (3) ◽  
pp. 463-468
Author(s):  
Yousif Alkhamees
Keyword(s):  
Maximal Ideal ◽  
Galois Ring ◽  
Primary Ring ◽  
Finite Rings ◽  
Zero Divisors ◽  
Special Case ◽  
Unique Maximal Ideal

According to general terminology, a ringRis completely primary if its set of zero divisorsJforms an ideal. LetRbe a finite completely primary ring. It is easy to establish thatJis the unique maximal ideal ofRandRhas a coefficient subringS(i.e.R/Jisomorphic toS/pS) which is a Galois ring. In this paper we give the construction of finite completely primary rings in which the product of any two zero divisors is inSand determine their enumeration. We also show that finite rings in which the product of any two zero divisors is a power of a fixed prime p are completely primary rings with eitherJ2=0or their coefficient subring isZ2nwithn=2or3. A special case of these rings is the class of finite rings, studied in [2], in which the product of any two zero divisors is zero.

scholarly journals Unit groups of finite rings with products of zero divisors in their coefficient subrings

2014 ◽  
Vol 95 (109) ◽  
pp. 215-220
Author(s):  
Chiteng’a Chikunji
Keyword(s):  
Finite Ring ◽  
Finite Rings ◽  
Group Of Units ◽  
Zero Divisors ◽  
Linearly Independent

Let R be a completely primary finite ring with identity 1 ? 0 in which the product of any two zero divisors lies in its coefficient subring. We determine the structure of the group of units GR of these rings in the case when R is commutative and in some particular cases, obtain the structure and linearly independent generators of GR.

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.

Dual Numbers and Topological Hjelmslev Planes

1983 ◽  
Vol 26 (3) ◽  
pp. 297-302 ◽  
Author(s):  
J. W. Lorimer
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Dual Numbers ◽  
Hjelmslev Plane ◽  
Product Topology ◽  
Topological Dimension ◽  
Locally Compact ◽  
Projective Hjelmslev Plane ◽  
Zero Divisors ◽  
Unique Maximal Ideal

AbstractIn 1929 J. Hjelmslev introduced a geometry over the dual numbers ℝ+tℝ with t2 = Q. The dual numbers form a Hjelmslev ring, that is a local ring whose (unique) maximal ideal is equal to the set of 2 sided zero divisors and whose ideals are totally ordered by inclusion. This paper first shows that if we endow the dual numbers with the product topology of ℝ2, then we obtain the only locally compact connected hausdorfT topological Hjelmslev ring of topological dimension two. From this fact we establish that Hjelmslev's original geometry, suitably topologized, is the only locally compact connected hausdorfr topological desarguesian projective Hjelmslev plane to topological dimension four.

scholarly journals -Regular Modules

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Areej M. Abduldaim ◽  
Sheng Chen
Keyword(s):  
Positive Integer ◽  
Maximal Ideal ◽  
Regular Ring ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Regular Module ◽  
Pure Submodules ◽  
Regular Rings

We introduced and studied -regular modules as a generalization of -regular rings to modules as well as regular modules (in the sense of Fieldhouse). An -module is called -regular if for each and , there exist and a positive integer such that . The notion of -pure submodules was introduced to generalize pure submodules and proved that an -module is -regular if and only if every submodule of is -pure iff   is a -regular -module for each maximal ideal of . Many characterizations and properties of -regular modules were given. An -module is -regular iff is a -regular ring for each iff is a -regular ring for finitely generated module . If is a -regular module, then .

scholarly journals NILPOTENCY OF THE GROUP OF UNITS OF A FINITE RING

2009 ◽  
Vol 79 (2) ◽  
pp. 177-182 ◽  
Author(s):  
DAVID DOLŽAN
Keyword(s):  
Nilpotent Group ◽  
Finite Ring ◽  
Finite Rings ◽  
Group Of Units

AbstractIn this paper we find all finite rings with a nilpotent group of units. It was thought that the answer to this was already given by McDonald in 1974, but as was shown by Groza in 1989, the conclusions that had been reached there do not hold. Here, we improve some results of Groza and describe the structure of an arbitrary finite ring with a nilpotent group of units, thus solving McDonald’s problem.

scholarly journals ON VARIETIES OF RINGS WHOSE FINITE RINGS ARE DETERMINED BY THEIR ZERO-DIVISOR GRAPHS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250019 ◽  
Author(s):  
A. S. Kuzmina ◽  
Yu. N. Maltsev
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge if and only if either xy = 0 or yx = 0. In the present paper, we study some properties of ring varieties where every finite ring is uniquely determined by its zero-divisor graph.

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