An addition theorem for the elementary Abelian group of type (p,p)

In this paper we prove that every element in the finite Abelian groupZ _p ×Z _p ,p>3,p prime, can be written as a sum over a subset of the setA, whereA is any set of non-zero elements ofZ _p ×Z _p with |A|=2p–2.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

Zero divisors and prime elements of bounded semirings

A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z（A） of nonzero zero divisors of A is A ／ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z（A） = A ／ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either ｜Z（A）｜ = 1 or ｜Z（A）｜ -- 2 and Z（A）2 ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I（R） is pure and shellable, where I（R） consists of all ideals of R.     

A decomposition theorem for numbers in which the summands have prescribed normality properties

If A is a set of integers, each exceeding unity, then every real number can be expressed as a sum of four numbers, each of which is non-normal with respect to every base belonging to A and is normal to every base which is not multiplicatively dependent on any element of A. This result is proved and generalized to allow noninteger bases.     

On quasi-injectivity and von Neumann regularity

The purpose of this note is to consider certain connections between injectivity,p-injectivity and a generalisation of quasi-injectivity notedGQ-injectivity (cf. definition below). It is proved that ifA is a leftGQ-injective ring andZ the left singular ideal ofA, thenA/Z is von Neumann regular andZ is the Jacobson radical ofA (this extends the well-known result ofY. Utumi for left continuous rings [9]). If the sum of any twoGQ-injective leftA-modules isGQ-injective, thenA is a left Noetherian, left hereditary, leftV-ring. Semi-prime rings whose faithful left modules areGQ-injective must be semi-simple Artinian. IfA is commutative, the following are equivalent: (1)A is a finite direct sum of field; (2) EveryGQ-injectiveA-module is injective; (3) AnyA-module isGQ-injective if, and only if, it isp-injective; (4) AnyA-module is quasi-injective if, and only if, it isp-injective. Also, a commutative ringA is hereditary Noetherian if, and only if, the sum of any twop-injectiveA-modules is injective.     

Translation invariance in groups of prime order

We prove that there is an absolute constant c>0 with the following property: if Z/pZ denotes the group of prime order p, and a subset A⊂Z/pZ satisfies 1<|A|<p/2, then for any positive integer there are at most 2m non-zero elements b∈Z/pZ with |(A+b)?A|?m. This (partially) extends onto prime-order groups the result, established earlier by S. Konyagin and the present author for the group of integers. We notice that if A⊂Z/pZ is an arithmetic progression and m<|A|<p/2, then there are exactly 2m non-zero elements b∈Z/pZ with |(A+b)?A|?m. Furthermore, the bound c|A|/ln|A| is best possible up to the value of the constant c. On the other hand, it is likely that the assumption can be dropped or substantially relaxed.     

Structure of large incomplete sets in abelian groups

Let G be a finite abelian group and A be a subset of G. We say that A is complete if every element of G can be represented as a sum of different elements of A. In this paper, we study the following question     

Additive properties of product sets in the field $$ \mathbb{F}_{p^2 } $$

For given positive integer n and ε > 0 we consider an arbitrary nonempty subset A of a field consisting of p ² elements such that its cardinality exceeds p ^2/n?ε . We study the possibility to represent an arbitrary element of the field as a sum of at most N(n, ε) elements from the nth degree of the set A. An upper estimate for the number N(n, ε) is obtained when it is possible.     

Partition theorems for Abelian groups

Let A be a finite matrix with integral entries and G be an Abelian group. Define A to be partition regular in G if for every partition of G/(0) into finitely many classes there exist elemens x₁,…,x_m contained in one class such that A(x₁,…,x_m)^T = 0. Theorem. A is partition regular in G iff at least one of the following statements holds. (i) There is x ∈ G/(0) such that A(x,…,x)^T = 0. (ii) A is partition regular in Z_p^?₀ (p prime) and Z_p^?₀ ? G. (iii) A is partition regular in Z and the set of orders of elements in G is unbounded.     

Torsion units in integral group rings of metacyclic groups

It is proved that if G is a split extension of a cyclic p-group by a cyclic p′-group with faithful action then any torsion unit of augmentation one of $\text{Z}$G is rationally conjugate to a group element. It is also proved that if G is a split extension of an abelian group A by an abelian group X with (|A|, |X|) = 1 then any torsion unit of $\text{Z}$G of augmentation one and order relatively prime to |A| is rationally conjugate to an element of X.     

Semiclean Rings

Abstract

The notion of semiclean elements in a ring is defined. Every clean element is semiclean. A ring R is said to be semiclean if every element in R is semiclean. The group ring Z _p G with G a cyclic group of order 3 is proved to be semiclean. The n × n matrix ring M _n (R) over a semiclean ring is semiclean. If R is a torsion free semiclean ring in which every element of R can be written as a sum of periodic and ±1, then R is clean. Every element in a semiclean ring R with 2 invertible is a sum of no more than 3 units.     

Class product and character product

For each finite group G, the product in the group ring of all the conjugacy class sums is a positive integer multiple of the sum of the elements in a special coset of the commutator subgroup G′, as Brauer and Wielandt first observed in the case G′ =  G. We show that the corresponding special element G! in A := G/G′ is the product of B! over specified subgroups B of A. Somewhat analogously, the product of all the irreducible characters of G, restricted to the center Z of G, is a multiple of a special linear character !G of Z, and !G is the product of !(Z/Y) over specified subgroups Y of Z.     

Minimal asymptotic bases for the natural numbers

The sequence A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be written as the sum of h elements of A. Let $\text{M}{}_{\mathrm{h}}{}^{\mathrm{A}}$ denote the set of elements that have more than one representation as a sum of h elements of A. It is proved that there exists an asymptotic basis A such that $\text{M}{}_{\mathrm{h}}{}^{\mathrm{A}}\text{(x) = O(x}{}^{\mathrm{<mtext>1?1</mtext><mtext>h+?</mtext>}}\text{)}$ for every ? > 0. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It is proved that there does not exist a sequence A that is simultaneously a minimal basis of orders 2, 3, and 4. Several open problems concerning minimal bases are also discussed.     

The critical number of finite abelian groups

Let G be an additive, finite abelian group. The critical number cr(G) of G is the smallest positive integer ? such that for every subset S⊂G?{0} with |S|?? the following holds: Every element of G can be written as a nonempty sum of distinct elements from S. The critical number was first studied by P. Erd?s and H. Heilbronn in 1964, and due to the contributions of many authors the value of cr(G) is known for all finite abelian groups G except for G≅Z/pqZ where p,q are primes such that . We determine that cr(G)=p+q−2 for such groups.     

On complete subsets of the cyclic group

A subset X of an abelian G is said to be complete if every element of G can be expressed as a nonempty sum of distinct elements from X.Let A⊂Zn be such that all the elements of A are coprime with n. Solving a conjecture of Erd?s and Heilbronn, Olson proved that A is complete if n is a prime and if . Recently Vu proved that there is an absolute constant c, such that for an arbitrary large n, A is complete if , and conjectured that 2 is essentially the right value of c.We show that A is complete if , thus proving the last conjecture.     

On sum sets of sidon sets,II

LetG be a finite abelian group,G?{Z _n, Z₂?Z_2n}. Then every sequenceA={g ₁,...,g_t} of $t = \frac{{4\left| G \right|}}{3} + 1$ elements fromG contains a subsequenceB?A, |G|=|G| such that $\sum\nolimits_{g_i \in B^{g_i } } { = 0 (in G)} $ . This bound, which is best possible, extends recent results of [1] and [22] concerning the celebrated theorem of Erdös-Ginzburg-Ziv [21].     

All-derivable points of operator algebras

Let A be an operator subalgebra in B(H), where H is a Hilbert space. We say that an element Z∈A is an all-derivable point of A for the norm-topology (strongly operator topology, etc.) if, every norm-topology (strongly operator topology, etc.) continuous derivable linear mapping φ at Z (i.e. φ(ST)=φ(S)T+Sφ(T) for any S,T∈A with ST=Z) is a derivation. In this paper, we show that every invertible operator in the nest algebra is an all-derivable point of the nest algebra for the strongly operator topology. We also prove that every nonzero element of the algebra of all 2×2 upper triangular matrixes is an all-derivable point of the algebra.     

Extension of continuous functions in digital spaces with the Khalimsky topology

The digital space Zn equipped with Efim Khalimsky's topology is a connected space. We study continuous functions Zn⊃A→Z, from a subset of Khalimsky n-space to the Khalimsky line. We give necessary and sufficient condition for such a function to be extendable to a continuous function Zn→Z. We classify the subsets A of the digital plane such that every continuous function A→Z can be extended to a continuous function on the whole plane.     

Nathanson heights in finite vector spaces

Let p be a prime, and let Zp denote the field of integers modulo p. The Nathanson height of a point is the sum of the least nonnegative integer representatives of its coordinates. The Nathanson height of a subspace is the least Nathanson height of any of its nonzero points. In this paper, we resolve a quantitative conjecture of Nathanson [M.B. Nathanson, Heights on the finite projective line, Int. J. Number Theory, in press], showing that on subspaces of of codimension one, the Nathanson height function can only take values about . We show this by proving a similar result for the coheight on subsets of Zp, where the coheight of A⊆Zp is the minimum number of times A must be added to itself so that the sum contains 0. We conjecture that the Nathanson height function has a similar constraint on its range regardless of the codimension, and produce some evidence that supports this conjecture.     

Torsion units in integral group rings of some metabelian groups,II

We prove that any torsion unit of the integral group ring ZG is rationally conjugate to a trivial unit if G = A ⋊ X with both A and X abelian, |Xz.sfnc; < p for every prime p dividing |A| provided either |X| is prime or A ic cyclic.