Title: on Maximal Substructures of Function Algebras with Composition

In this talk the idea of an intrinsic extension of a ring, first proposed by Faith and Utumi, is generalized and studied in its own right. For these types of ring extensions, it is shown that, with relatively mild conditions on the base ring, R, a complete set of primitive idempotents (a complete set of left triangulating idempotents, a complete set of centrally primitive idempotents) can be constructed for an intrinsic extension, T , from a corresponding set in the base ring R. Examples and applications are given for rings that occur in Functional Analysis. Furthermore, it is shown that our main results provide a new method for attempting to investigate the well-known Zero Divisor Problem in Group Ring Theory and possibly extend partial solutions to the problem. Kent Neuerburg, Southeastern Louisiana University, USA Title: Simple Rings and Covered Groups Abstract: Let (G,+) be a group and C = {C1, . . . , Cn} be a collection of abelian subgroups that forms a cover of G. We consider the set of functions R(C) = {f : G→ G | f ∈ End(Ci)∀Ci ∈ C}. We note that (R(C),+, ◦) forms a ring in which the operations are point-wise addition and function composition. We consider relationships between the group G and the ring R(C). In particular, we will focus on simple rings. Guenter Pilz, Johannes Keplar Universität, Austria Title: Nearrings and Algebraic Equations Abstract: What is an “equation”? The best answer: a pair (p, q) of polynomials. A solution of this equation is an element r such that p ◦ r = q ◦ r, where ◦ denotes composition. So we are fully in the polynomial NEAR-ring rather than in polynomial rings. And from there, it is quite clear how to define equations in groups, modules, non-commutative rings, and the like. The criterion when an equation (or a system of equations) has a solution turns out to have a “near-ringish flavor” as well. Much of the presented material comes from joint work with Erhard Aichinger. Bhavanari Satyanarayana, Acharya Nagarjuna University, India Title: Prime Graph of a Nearring Abstract: We define a new concept called “Prime Graph of a nearring N (denoted by PG(N))”. We make observations, present examples and prove results which explain certain relations between prime near-rings and prime graphs. This paper forms a new bridge between “Graph Theor” and the algebraic concept “Nearrings”. We consider prime nearrings. We obtained two equivalent conditions (in terms of We define a new concept called “Prime Graph of a nearring N (denoted by PG(N))”. We make observations, present examples and prove results which explain certain relations between prime near-rings and prime graphs. This paper forms a new bridge between “Graph Theor” and the algebraic concept “Nearrings”. We consider prime nearrings. We obtained two equivalent conditions (in terms of