Volume 2, Issue 1

In 1964, Bertram Yood posed the following problem: whether the intersection of all closed maximal regular left ideals of a topological ring coincides with the intersection of all closed maximal regular right ideals of this ring. It is proved that these two intersections coincide for advertive and simplicial topological rings and, using this result, it is shown that the topological left radical and the topological right radical for every advertive and simplicial topological algebra coincide.

We prove that a separable Hausdor_ topological space X containing a cocountable subset homeomorphic to [0, ω 1 ] admits no separately continuous mean operation and no diagonally continuous n-mean for n ≥ 2.

Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if forevery neighbourhood V of the identity e in G , there exists a natural number n such that G = V n (resp.we have G = ∪ n ∈N V n ). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded)premeager paratopological group is a topological group. Also,we prove that if G is a totally ω-pseudoboundedparatopological group such that G is a Lusin space, then is G a topological group. We present some examplesof paratopological groups with interesting properties: (1) There exists a metrizable, zero-dimensional and pseudobounded topological group; (2) There exists a Hausdorff ω-pseudobounded paratopological group G such that G contains a dense subgroupwhich is not ω-pseudobounded; (3) There exists a Hausdorff connected paratopological group which is not ω-pseudobounded.

A topological algebra A is said to be fundamental if there exists b > 1 such that for every sequence( x n ) in A , ( x n ) is Cauchy whenever the sequence b n ( x n − x n-1 ) tends to zero as n → ∞. Let A be a complex unital fundamental F -algebra with bounded elements such that A * separates the points on A . Then we prove that the spectrum σ(a) of every element a ∈ A is nonempty compact. Moreover, if A is a division algebra,then A is isomorphic to the complex numbers ℂ. This result is a generalization of Gelfand-Mazur theoremfor a large class of F -algebras, containing both locally bounded algebras and locally convex algebras withbounded elements.

The aim of this paper is to consider the ringswhich can be graded by completely simple semigroups.We show that each G-graded ring has an orthonormal basis, where G is a completely simple semigroup. Weprove that if I is a complete homogeneous ideal of a G-graded ring R, then R/I is a G-graded ring.We deducea characterization of the maximal ideals of a G-graded ring which are homogeneous. We also prove that if Ris a Noetherian graded ring, then each summand of it is also a Noetherian module..