θ*-Relation on Hypermodules and Fundamental Modules Over Commutative Fundamental Rings

Abstract

The main tools in the theory of hyperstructues are the fundamental relations. The fundamental relation on a hypermodule over a hyperring was already introduced by Vougiouklis. The fundamental relation on a hypermodule over a hyperring is defined as the smallest equivalence relation so that the quotient would be the module over a ring. Note that generally the commutativity with respect to both sum in the (fundemental) module and product in the (fundamental) ring are not assumed. In this article we introduce a new strongly regular equivalence relation on hypermodules so that the quotient is module (with abelin group) over a commutative ring. Also we state the conditions that is equivalent with the transitivity of this relation and finally we characterize the complete hypermodules over hyperrings.