A hyperstructural approach to essentiality

A Hyperstructural Approach to Semisimplicity

“…Let M be an R-hypermodule. In [7], a subhypermodule N of M is called essential in M and denoted by Let M and N be two hypermodules over a hyperring R. The set of all normal homomorphisms from M to N is denoted by Hom R (M, N). For f , g ∈ Hom R (M, N) and m ∈ M, as in [37], define…”

“…Krasner [5] solves a problem in the approximation of a complete valued field by a sequence of such fields by using hyperfields. Recently, as more general structures of Krasner hyperrings and Krasner hyperfields, these notions of general hyperrings and general hypermodules have been introduced and studied by many authors in a series of papers [6][7][8][9]. Let p ∈ P, where P is the set of all positive prime integers.…”

t-extending Krasner hypermodules

“…Let M and N be two R-hypermodules. Recall from [9] that the function f : M −→ N is called a homomorphism if f (x + y) ⊆ f (x) + f (y) and f (x.r) = f (x).r for all x, y ∈ M and r ∈ R. Also, f is said to be a strong homomorphism if f (x + y) = f (x) + f (y) and f (x.r) = f (x).r for all x, y ∈ M and r ∈ R. Note that in this case, f (0 M ) = 0 N . If a strong homomorphism f is one-to-one and surjective function, it is called a strong isomorphism.…”