0. Introduction. 1. Let G be a group of linear transformations on a finite dimensional real or complex vector space X. Assume X is completely reducible as a G-module. Let 5 be the ring of all complexvalued polynomials on X, regarded as a G-module in the obvious way, and let J C 5 be the subring of all G-invariant polynomials on X. Now let J be the set of all ƒ £ J having zero constant term and ...

Let L denote the free Lie ring of rank 2r with free generators xi , yi (1 i r), and let τ be the automorphism of L that interchanges xi and yi for all i . We construct a Z-basis of L that consists of pairs of elements which are interchanged by τ , and elements upon which τ acts as multiplication by −1.

The main goal of this paper is to study the class countably $\mathcal {A}$-rings (or McCoy rings) introduced by T. Lucas in [The diameter a zero divisor graph, J. Algebra 301, 174-193, 2006] which turns out lie properly between $ \mathcal{A}$-rings and total-$\mathcal{A}$-rings. Also, we introduce investigate module theoretic version {A}$-ring notion, namely {A}$-modules. Our focus shed on beha...

To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U − q (g), where g is the Kac-Moody Lie algebra associated with the graph.