Hopkins-Levitzki theorem for Krasner hyperrings

Abstract

It is well known that Hopkins-Levitzki theorem connected the d.c.c.
and a.c.c. in modules over semisimple rings, in particular, every right
Artinian ring is right Noetherian. This theorem was rst formulated
by C. Hopkins and J. Letvitzki in 1933. It is noted that W. Krull
and Y. Akizuki developed the general theory relating the structures
of Artinian rings and Noetherian rings around 1940. A special case
of this type is the hyperring introduced by Krasner Hyperstructures
represent a natural extension of classical algebraic structures and they
were introduced in 1934 by the French mathematician F. Marty. The
hyperrings is a structure that satises the ring-like axioms and special
case of this type is the hyperring such that introduced by Krasner. In
this paper, our aim is to generalize and extend this celebrated Hopkins-
Levitzki theorem from non-commutative rings to Krasner hyperring.
Also, we prove that a Krasner hyperring R is Noetherian if and only
if it satises the ascending chain conditions of prime hyperideals.