On the multiplicative structure of finite division rings

All division ringsD of 16, 27, 32, 125 and 343 elements are shown to have aright primitive elementp such that

An example of two von Neumann regular rings with nonisomorphic morita equivalent multiplicative semigroups

The aim of this note is to construct an example of two von Neumann regular ringsR andS such that their multiplicative semigroups (R,.) and (S,.) are Morita equivalent but nonisomorphic.     

Cohomology rings of multiplicative fibrations

Let p be an odd prime. Let be the fibre space induced from an H-map where K is a generalized Eilenberg MacLane space and X is a simply connected H-space. Such spaces occur frequently in Postnikov towers and connective covers. In this paper, we compute the mod p cohomology of as a ring. The ring depends on the structure of imf^* and the structure of subkerf^* as modules over the Steenrod algebra.     

Homology of multiplicative Lie rings

Two homology theories of multiplicative Lie rings are constructed, studied and compared with the usual homology theories of groups and Lie rings. Central extensions of multiplicative Lie rings are introduced. It is shown that the Steinberg multiplicative Lie ring of a ring is the direct product of the Steinberg group (viewed as a multiplicative Lie ring under the commutator bracket) and the Steinberg Lie ring.     

Arithmetical rings and multiplicative lattices

Summary Arithmetical rings and, in particular, Prüfer domains, are characterized in terms of the principal elements of their lattices of ideals. A study is made of certain multiplicative lattices which have properties in common with the lattices of ideals of arithmetical rings. Entrata in Redazione il 15 novembre 1968.     

Completely O-simple multiplicative ideals of rings

The purpose of this note is the characterization of rings which are generated by a completely O-simple multiplicative ideal. It will follow as a corollary of this characterization that a ring which is multiplicatively completely O-simple is a division ring [4].     

Abundant semigroups with a multiplicative quasi-adequate transversal

The concept of quasi-adequate transversals is introduced. Some properties and characterizations for abundant semigroups with a multiplicative quasi-adequate transversal are obtained. In particular, a structure theorem for such abundant semigroups is given. Finally, some special cases are considered. This research was partially supported by the National Natural Science Foundation of China (No. 10571077) and the Natural Science Foundation of Gansu Province (No. 3ZS052-A25-017).     

Arithmetic of semigroups of series in multiplicative systems

We study the arithmetic of a semigroup M_P\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?_{n = 0}^￥ a_nc_n(x)    ( a_n 3 0,?_{n = 0}^￥ a_n = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { c_n }_{n = 0}^￥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup M_P\mathcal{M}_{\mathcal{P}}, analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R ⁿ are true. We describe the class I₀(M_P)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in M_P\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence.     

On 2-absorbing commutative semigroups and their applications to rings

A commutative ring R is called 2-absorbing (Badawi in Bull. Aust. Math. Soc. 75:417–429, 2007) if for arbitrary elements a,b,c∈R, abc=0 if and only if ab=0 or bc=0 or ac=0. In this paper we study this concept in a more general framework of commutative (multiplicative) semigroups with 0. The results obtained apply to many ring theoretic situations and make it possible to describe similarities and differences among some variants of the notion. We pay a particular attention to graded rings. We also show that a conjecture from (Anderson and Badawi in Commun. Algebra 39:1646–1672, 2011) concerning n-absorbing rings holds for rings with torsion-free additive groups.