A Hyperstructural Approach to Semisimplicity

Abstract

In this paper, we provide the basic properties of (semi)simple hypermodules. We show that if a hypermodule M is simple, then (End(M),·) is a group, where End(M) is the set of all normal endomorphisms of M. We prove that every simple hypermodule is normal projective with a zero singular subhypermodule. We also show that the class of semisimple hypermodules is closed under internal direct sums, factor hypermodules, and subhypermodules. In particular, we give a characterization of internal direct sums of subhypermodules of a hypermodule.

A Characterization of Normal Injective and Normal Projective Hypermodules

“…be a short exact sequence of hypermodules. Since M is semisimple, it follows from [24], Theorem 9, that Ker(ϕ) = Im(ψ) is a direct summand of M. From this, the short exact sequence E is splitting.…”

“…(3) ⇒ (4) Let M be any left R hypermodule and U ⊆ M. Applying (3), we obtain that U is normal injective and, so, from Corollary 3, M has the decomposition M = U ⊕ V, where V is a subhypermodule of M. Hence, M is semisimple according to [24], Theorem 9.…”

“…(7) ⇒ (1) From [24], Corollary 4, it is enough to show that whenever I ⊵ R, I = R. Let us assume the opposite, that is, I ̸ = R. Therefore, there is a maximal left hyperideal P of R such that P contains the essential left hyperideal I. Applying [24], Theorem 3, R P is a simple R hypermodule. Now, from (7)…”

“…Proof. Owing to the fact that M is a semisimple hypermodule, it follows from [24], Theorem 9, that Im( f ) is a direct summand of M. Therefore, we can write M = Im( f ) ⊕ M 1 for some subhypermodule M 1 of M. Note that λ = f −1 : Im( f ) −→ U is a normal isomorphism. Set h = gλπ, where π : M −→ Im( f ) is the canonical projection.…”