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.

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.

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 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 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.

scholarly journals Finite unitary rings with a single subgroup of prime order of the group of units

2020 ◽  
pp. 2150234
Author(s):  
Mostafa Amini ◽  
Mohsen Amiri
Keyword(s):  
Prime Number ◽  
Prime Order ◽  
Finite Ring ◽  
Group Of Units

Let [Formula: see text] be a unitary ring of finite cardinality [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text]. We show that if the group of units of [Formula: see text] has at most one subgroup of order [Formula: see text], then [Formula: see text] where [Formula: see text] is a finite ring of order [Formula: see text] and [Formula: see text] is a ring of cardinality [Formula: see text] which is one of the six explicitly described types.

scholarly journals Group of units in a finite ring

2002 ◽  
Vol 170 (2-3) ◽  
pp. 175-183 ◽  
Author(s):  
David Dolz̆an
Keyword(s):  
Finite Ring ◽  
Group Of Units

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.

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