we show  that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. as aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. this generalizes and simplifies a result of  dung and   smith. as another consequen...

This section define a level subring or level ideals obtain a set of necessary and sufficient condition for the equality of two ideals and characterizes field in terms of its fuzzy ideals. It also presents a procedure to construct a fuzzy subrings (fuzzy ideals) from any given ascending chain of subring ideal. We prove that the lattice of fuzzy congruence of group G (respectively ring R) is isom...

Received: March 21, 2011 Accepted: April 14, 2011 doi:10.5539/jmr.v3n3p89 Abstract It is shown that if a group ring satisfies the idempotent conjecture and some other restrictions it will be representational consistent. As an application we show the real numbers as a field are representational consistent. This leads to the problem of whether a circle can be given a field structure that can be s...

It is then easily verified that RG satisfies the ring axioms; in fact, RG is a linear algebra over R. (We write all groups multiplicatively, and denote group identities by 1; we also use 1 for the unit element of R if there is one.) If R, in addition to being a ring, is a Banach algebra (i.e., an algebra over the complex field K, with a submultiplicative norm which makes R a Banach space), then...

An element a in a ring R is called left morphic if there exists b ∈ R such that 1R(a)= Rb and 1R(b)= Ra. R is called left morphic if every element ofR is left morphic. An element a in a ring R is called left π-morphic (resp., left G-morphic) if there exists a positive integer n such that an (resp., an with an = 0) is left morphic. R is called left π-morphic (resp., left G-morphic) if every elem...

We give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by Cohen and Montgomery in [1]. This duality is then used to extend known results on skew group rings to corresponding results for large classes of group-graded rings. Finally we modify the construction slightly to handle infinite groups. Introduction. In the first section...

This paper consists of two parts. The first is concerned with free products in linear groups and uses the usual “ping pong” lemma and attractors to prove the results. What is new here is that we allow certain subspaces of V associated with the semisimple and generalized transvection operators to have dimensions larger than 1. The second part is concerned with applications of this machinery to i...

In this brief note, we study algebraic elements in the complex group algebra C[G]. Specifically, suppose £ e C[G] satisfies /(£) = 0 for some nonzero polynomial f(x) e C[x]. Then we show that a certain fairly natural function of the coefficients of Z is bounded in terms of the complex roots of f(x). For G finite, this is a recent observation of [HLP], Thus the main thrust here concerns infinite...

While we were graduate students, Marty Isaacs and I worked together on the character theory of finite groups, studying in particular the character degrees of finite p-groups. Somewhat later, my interests turned to ring theory and infinite group theory. On the other hand, Marty continued with character theory and soon became a leader in the field. Indeed, he has had a superb career as a research...

In [5] and [6], a nil clean ring was defined as a ring for which every element is the sum of a nilpotent and an idempotent. In this short article we characterize nil clean commutative group rings.