We show that cohopfian modules with finite exchange have countable exchange. In particular, a module whose endomorphism ring is Dedekind-finite and π-regular has the countable exchange property. We also show that a module whose en-domorphism ring is Dedekind-finite and regular has full exchange. Finally, working modulo the Jacobson radical, we prove that any module with the (C 2) property and a...

nitrate is an anion of major importance particularly in biological areas. excessive amounts in water supplies indicate pollution from sewage or agricultural effluents. the effects of excessive quantities of nitrate in water are well known. in this research structural modification of natural clinoptilolite by isomorphic substitution was performed. isomorphously substituted zeolite was prepared u...

A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U . A subset W is called a resolving set if every vertex in G is uniquely determined by its distances to the vertices of W . Determining (resolving) sets are said to have the exchange property in G if whenever S and R are minimal determining (resolvi...

In this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $Z_{2}$. We also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.

Photocatalyst nanocomposites of ZnO/bentonite clay are synthesized by Solid-state ion exchange method. Ion exchange intercalation process of clays is used to incorporate the catalyst into the basal space of the layered structure of clays. The purpose of this study is to find a new method, which is focused on simplifying and saving time to prepare ZnO-bentonite composite with photocatalyst prope...

This paper proposes an original exchange property of valuations.This property is shown to be equivalent to a property described by Dress and Terhalle in the context of discrete optimization and matroids and shown there to characterize the valuations for which the demand oracle can be implemented by a greedy algorithm. The same exchange property is also equivalent to a property described indepen...