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Based on the works of Axtell et al., Anderson et al., and Ghanem on associate, domainlike, and presimplifiable rings, we introduce new hyperrings called associate, hyperdomainlike, and presimplifiable hyperrings. Some elementary properties of these new hyperrings and their relationships are presented.

The study of strongly associate rings began with Kaplansky in [

The theory of hyperstructures was introduced in 1934 by Marty [

Hyperrings are essentially rings with approximately modified axioms. Hyperrings

In this paper, we present and study associate, hyperdomainlike, and presimplifiable hyperrings. The relationships between these new hyperrings are presented.

In this section, we will provide some definitions that will be used in the sequel. For full details about associate, domainlike, and presimplifiable rings, the reader should see [

Let

Integral domains, domainlike, and local rings are presimplifiable rings and hence are associate rings.

It is easy to check that

Let

A hypergroupoid

A hypergroupoid

Let

there exists a neutral element

for every

Let

multiplication

If

A hyperring

For any

Throughout this section, all hyperrings will be assumed to be commutative Krasner hyperrings with unity.

Let

Let

if

Let

Let

(1) and (2) are clear. For (3), suppose that

If

A hyperring is a

The following example is presented in [

Let

It is easy to see that Example

Let

If

(1) and (2) are obvious. For (3), suppose that

Let

The hyperring in Example

Let

Let

Let

for

(

(

(

(

(

(

(

Let

Suppose that

Let

It follows from Theorem

Any hyperdomain or any quasilocal hyperring is an associate hyperring.

This is immediate from Theorems

Let

(

(

(3) implies (1): suppose that

Let

Let

If

If

If

A direct product of superassociate hyperrings need not be superassociate.

See Anderson et al. [

Let

The proof is similar to the proof of Theorem 2.1 in [

Let

Suppose that

Conversely, suppose that

Let

The proof follows from Theorem

Let

Suppose that

Let

Suppose that

Let

Suppose that

Conversely, suppose that

Let

The proof is similar to the proof of Theorem

Let

The proof is similar to the classical ring and therefore omitted.

Let

Suppose that

Conversely, suppose that

Let

Let

Every hyperideal of a presimplifiable hyperring

It is obvious.

Let

Suppose that

Conversely, suppose that

Let

Suppose that

Conversely, suppose that

Let

A strong homomorphism

If

Let

Let

Let

The proof is similar to the proof of Proposition 6 in [

Let

The proof is similar to the proof of Theorem 7 in [

Let

The proof is similar to the proof of Theorem 2.3 in [

Let

The proof is similar to the proof of Theorem 2.5 in [

Let

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the referees for their critical reading of the paper and their suggestions.