### 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.

### Refbacks

- There are currently no refbacks.