R-parts and modules derived from strongly U-regular relations on hypermodules

This paper concerns a new relationship between hypermodules  and modules. We generalize the notion of complete parts and $\theta$-parts by the notion of $\Re$-parts on hypermodulesand then $\Re$-closures of hypermodules  as a generalization of $\theta$-closures  are defined. In addition,we give the notion of a strongly $\mathcal{U}$-regular relation on hypermodules and investigate some properties of it

On the heart of hypermodules

This paper presents some types of hypermodules, associated to an arbitrary hypermodule, studying properties and characterizing their hearts. Also, we establish a few results concerning the sequence of heart, which can be associated to a hypermodule, in connection with subhypermodules generated by a non-empty set, by a union of subhypermodules or by the intersection of subhypermodules. Finally, we study several properties of $1$-hypermodules

Commutative fundamental (m, n)-modules

In this paper, we introduce the concept of fundamental relation θ∗ on an (m, n)-hypermodule M as the smallest equivalence relation such that M/θ∗ is a commutative (m, n)-module, and then some related properties are investigated