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 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.

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 The n-insertive subgroups of units

2007 ◽  
Vol 75 (1) ◽  
pp. 23-26
Author(s):  
David Dolžn
Keyword(s):  
Local Ring ◽  
Jacobson Radical ◽  
Finite Ring ◽  
Local Rings ◽  
Arbitrary Integer ◽  
Group Of Units ◽  
Direct Sum ◽  
Group 1

Let R be a finite ring. Let us denote its group of units by G = G(R) and its Jacobson radical by J = J(R). Let n be an arbitrary integer. We prove that R is an n-insertive ring if and only if G is an n-insertive group and show that every n-insertive finite ring is a direct sum of local rings. We prove that if n is a unit, then the local ring R is n-insertive if and only if its Jacobson group 1 + J is n-insertive and find an example to show that this is not true if n is a non-unit.

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 Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 307
Author(s):  
Sami Alabiad ◽  
Yousef Alkhamees
Keyword(s):  
Unit Group ◽  
Galois Ring ◽  
Direct Decomposition ◽  
Finite Ring ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Group Of Units ◽  
Lattice Of Ideals

A finite ring with an identity whose lattice of ideals forms a unique chain is called a finite chain ring. Let R be a commutative chain ring with invariants p,n,r,k,m. It is known that R is an Eisenstein extension of degree k of a Galois ring S=GR(pn,r). If p−1 does not divide k, the structure of the unit group U(R) is known. The case (p−1)∣k was partially considered by M. Luis (1991) by providing counterexamples demonstrated that the results of Ayoub failed to capture the direct decomposition of U(R). In this article, we manage to determine the structure of U(R) when (p−1)∣k by fixing Ayoub’s approach. We also sharpen our results by introducing a system of generators for the unit group and enumerating the generators of the same order.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document