Rings consisting entirely of certain elements

We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; Z₃ ⊕ Z₃; Z₃ ⊕ B where B is a Boolean ring; local ring with nil Jacobson radical; M₂(Z₂) or M₂(Z₃); or the ring of a Morita context with zero pairings where the underlying rings are Z₂ or Z₃.     

The commuting graphs of some subsets in the quaternion algebra over the ring of integers modulo <Emphasis Type="Italic">N</Emphasis>

Let R be an arbitrary ring, S be a subset of R, and Z(S) = {s ∈ S | sx = xs for every x ∈ S}. The commuting graph of S, denoted by Γ(S), is the graph with vertex set S \ Z(S) such that two different vertices x and y are adjacent if and only if xy = yx. In this paper, let I _n , N _n be the sets of all idempotents, nilpotent elements in the quaternion algebra ℤ _n [i, j, k], respectively. We completely determine Γ(I _n ) and Γ(N _n ). Moreover, it is proved that for n ≥ 2, Γ(I _n ) is connected if and only if n has at least two odd prime factors, while Γ(N _n ) is connected if and only if n ∈ 2, 2², p, 2p for all odd primes p.     

Walker's Cancellation Theorem

Walker's cancellation theorem says that, if B ⊕ Z is isomorphic to C ⊕ Z in the category of abelian groups, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the original theorem does not have a constructive proof even if B and C are subgroups of the free abelian group on two generators. Both of these results contrast with a group whose endomorphism ring has stable range one, which allows a constructive proof of cancellation and also a proof in any diagram category.     

Z-cyclic triplewhist tournaments—The noncompatible case,part I

Two odd primes odd, are said to be noncompatible if b₁ ≠ b₂. For all noncompatible (ordered) pairs of primes (p₁, p₂) such that p_i ≡ p_i < 200, i = 1,2 we establish the existence of Z-cyclic triplewhist tournaments on 3p₁ p₂ + 1 players. It is believed that these results are the first examples of such tournaments, indeed the first examples of Z-cyclic whist tournaments for such players. In Part 2 we extend the results of this study and establish the existence of Z-cyclic triplewhist tournaments on players for all α₁ ≥ 1, α₂ ≥ 1 and p₁, p₂ as described above. © 1997 John Wiley & Sons, Inc.     

T-Semisimple Modules and T-Semisimple Rings

We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z ₂(M) ⊕ S(M) (where Z ₂(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R _R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions.     

The unique rank property among gabriel localizations

It is not uncommon for rings to have Gabriel localizations which do not possess the unique rank (UR) property although the rings themselves do have UR. We show that if F is a Gabriel filter of right ideals on a ring R and R_F is the corresponding Gabriel localization, then free R_F?modules of ranks m and n are isomorphic if and only if some F-dense submodule of (R/T_f(R))^m is isomorphic to some F-dense submodule of (R/T_F(R))ⁿ, where T_F(R) is the F-torsion ideal of R.     

Certain decompositions of matrices over Abelian rings

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ?. We prove that M ⁿ (R) is nil clean if and only if R/J(R) is Boolean and M _n (J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ?₃, B or ?₃ ⊕ B where B is a Boolean ring, and that M _n (R) is weakly nil clean if and only if M _n (R) is nil clean for all n ≥ 2.     

On the existence of semifields of prime power order admitting free automorphism groups

The purpose of this paper is to prove the existence of semifields of order q ⁴ for any odd prime power q = p^r, q > 3, admitting a free automorphism group isomorphic to Z ₂ × Z ₂.     

On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.     

On endomorphism-regularity of zero-divisor graphs

The paper studies the following question: Given a ring R, when does the zero-divisor graph Γ(R) have a regular endomorphism monoid? We prove if R contains at least one nontrivial idempotent, then Γ(R) has a regular endomorphism monoid if and only if R is isomorphic to one of the following rings: Z₂×Z₂×Z₂; Z₂×Z₄; Z₂×(Z₂[x]/(x²)); F₁×F₂, where F₁,F₂ are fields. In addition, we determine all positive integers n for which Γ(Z_n) has the property.     

Looking for difference sets in D2p × Zq

We consider groups D_2p × Z_q, with p and q odd primes, q < p, and for which each prime dividing n has order p − 1 (mod p). If such a group contains a nontrivial difference set, D, our main theorem gives constraints on the parameters of D. This in turn rules out difference sets in some groups of this form. For instance, D₂₂ × Z₃ contains no nontrivial difference set. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 35–41, 2000     

Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.     

Homomorphisms between A-Projective Abelian Groups and Left Kasch-Rings

Glaz and Wickless introduced the class G of mixed abelian groups A which have finite torsion-free rank and satisfy the following three properties: i) A _p is finite for all primes p, ii) A is isomorphic to a pure subgroup of _P A _P and iii) Hom(A, tA) is torsion. A ring R is a left Kasch ring if every proper right ideal of R has a non-zero left annihilator. We characterize the elements A of G such that E(A)/tE(A) is a left Kasch ring, and discuss related results.     

On Finite Groups in Which Cyclic Subgroups of the Same Order are Conjugate

We consider finite groups G for which any two cyclic subgroups of the same order are conjugate in G. We prove various structure results and, in particular, that any such group has at most one non-abelian composition factor, and this is isomorphic to PSL(2, p ^m ), with m odd if p is odd, or to Sz(2^2m+1), or to one of the sporadic groups M ₁₁, M ₂₃, or J ₁.     

NILPOTENT GROUPS AND ABELIANIZATION

We study the question of what properties of nilpotent groups are shared by their abelianizations. We identify two such properties—that of being a π-torsion group, where π is a family of primes, and that of having q^th roots, for some prime q. We use these properties to provide simplified proofs of the following theorems in the localization of nilpotent groups.

Let H, K be subgroups of the nilpotent group N and let P be a family of primes. Then [H, K] P = [H_P, K_p]

Let the group G act on the nilpotent group N. Then G acts compatibly on N_p and (Γ^G _i N)_P = Γ^G _i(N_p).

The second theorem above is then applied to the study of the localization of relative groups, in the sense of [4].     

Catenarian Numbers

An integer n is called catenarian if, whenever L/K is an n-dimensional field extension, all maximal chains of fields going from K to L have the same length. Catenarian field extensions and catenarian groups are defined analogously. If n is an even positive integer, 6n is non-catenarian. If n ≥ 3 is odd, there exist infinitely many odd primes p such that p ² n is non-catenarian. A finite-dimensional field extension is catenarian iff its maximal separable subextension is. If q < p are odd primes where q divides p ? 1 (resp., q divides p + 1), every (resp., not every) group of order p ² q is catenarian.     

Boolean Algebras,Generalized Abelian Rings,and Grothendieck Groups

ABSTRACT

A ring R is called generalized Abelian if for each idempotent e in R, eR and (1 ? e)R have no isomorphic nonzero summands. The class of generalized Abelian rings properly contains the class of Abelian rings. We denote by GAERS ? 1 the class of generalized Abelian exchange rings with stable range 1. In this article we prove, by introducing Boolean algebras, that for any R ∈ GAERS ? 1, the Grothendieck group K ₀(R) is always an Archimedean lattice-ordered group, and hence is torsion free and unperforated, which generalizes the corresponding results of Abelian exchange rings. Our main technical tool is the use of the ordered structure of K ₀(R)⁺, which provides a new method in the study of Grothendieck groups.     

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups III

Let p₁, p₂, p₃ be primes. This is the final paper in a series of three on the (p₁, p₂, p₃)-generation of the finite projective special unitary and linear groups PSU ₃(pⁿ), PSL ₃(pⁿ), where we say a noncyclic group is (p₁, p₂, p₃)-generated if it is a homomorphic image of the triangle group T_{p1, p2, p3} . This article is concerned with the case where p₁ = 2 and p₂ ≠ p₃. We determine for any primes p₂ ≠ p₃ the prime powers pⁿ such that PSU ₃(pⁿ) (respectively, PSL ₃(pⁿ)) is a quotient of T = T_{2, p2, p3} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU ₃(pⁿ)) (respectively, Hom(T, PSL ₃(pⁿ))) is surjective as pⁿ tends to infinity.