r. ameri

In this paper, we consider a hypervector space (in the sense of Tallini) $V$ over a field $K$. We use the fundamental relation $\varepsilon^*$ over $V$, as the smallest equivalence relation on $V$, to derived the fundamental vector space ${V}/\varepsilon^*$. In this regards, we prove that if $V$ is a (resp. quasi) topological hypervector space, then the fundamental vector space ${V}/\varepsilon^*$ with the property that each open subset of it is a complete part, then its fundamental vector space ${V}/\varepsilon^*$ is a topological vector space. Finally, we prove that for a topological vector space $(V,+,\cdot,K)$ and every subspace $W$ of $V$, the hypervector space $(\overline{V},+,\circ,K)$ is a topological hypervector space and we will prove $\overline{V}/\varepsilon^*$ and $V/W$ are homeomorphic, where $\overline{V}=V$.

We introduce a new regular relation δ on a given group G and show that δ is a congurence relation on G, with respect to  module the commutator subgroup of G. Then we show that the effect of this relation on the fundamental relation β is equal to the fundamental relation γ, and we conclude that, if ρ is an arbitrary strongly regular relation on the hypergroup  H, then the effect of δ on ρ, results in a strongly regular relation on H such that its quotient is an abelian group.

In this paper, we construct the concept of general  Krasner  hyperring based on the  ring  structures and the left general Krasner hypermodule based on the  module structures.  This study introduces  the  trivial left general Krasner hypermodules and  proves that the  trivial left general Krasner   hypermodules  are  different from left  Krasner   hypermodules. We show that for any given general  Krasner  hyperring $R$ and trivial left general Krasner   hypermodules $A, B, {bf_{R}h}$om$(A, B)$ is a left general Krasner   hypermodule and  ${bf_{R}h}$om$(-, B)$,     $ ({bf_{R}h}$om$(A, -) )$ is an   exact covariant functor (contravariant). Finally, we  show that the category ${bf_{R}GKH}$mod (left  trivial general Krasner hypermodules and all (homomorphisms) is an abelian  category and  trivial left general Krasner hypermodules have  a normal injective resolution.

 This paper extends multirings to a novel concept as general multirings, investigates their properties and presents a special general multirings as notation of (m; n)-potent general multirings. This study analyzes the differences between class of multirings, general multirings and general hyperrings and constructs the class of (in)finite general multirings based on any given non-empty set. In final, we define the concept of hyperideals in general multirings and compare with hyperideals in othersimilar (hyper)structures.