Composition hyperrings

Abstract

In this paper we introduce the notion of composition hyperring. We show that the composition structure of a composition hyperring is determined by a class of its strong multiendomorphisms. Finally, the three isomorphism theorems of ring theory are derived in the context of composition hyperrings.

A note on composition (m, n)-hyperrings

“…Based on the notion of composition rings [1] and taking into account the properties of the hyperrings of polynomials [13], a new type of hyperrings, called composition hyperrings, has been introduced in [7]. Following the same idea, in this note we have investigated the properties of composition (m, n)-hyperrings, emphasizing the relations between them and the composition hyperrings, connection illustrated by several examples.…”

“…For m = n = 2 we get that (R, f, g, •) is a composition hyperring, defined in [7]. For this reason throughout this paper, when we talk about a composition (m, n)-hyperring we intend (m, n) = (2, 2).…”

“…Now, similar to Example 3.10 in [7], define two functions h : R −→ P * (R) and l : R −→ P * (R) by h(x) = A · x = {2 q · x | q ∈ Q} and l(x) = −A·x = {−2 q ·x | q ∈ Q}. Obviously, h and l are strong endomorphisms of (R, f, g).…”

“…Define the hyperoperations ⊕ A and A on R as x ⊕ A y = xA + yA and x A y = xAy. By Example 3.10 in [7], (R, ⊕ A , A ) is a commutative hyperring. Now, we define the m-ary and n-ary hyperoperations "f " and "g" on R as follows:…”

“…In the first work on this topic, Cristea and Jančić-Rašović [7] defined and studied the composition hyperrings, emphasizing their interesting properties in relation with endomorphisms of hyperrings. This work can be extended to other two directions: the first one, the topic of this note, deals with composition (m, n)-hyperrings, while the second one (subject investigated in [21]) with n-ary composition hyperrings, i.e.…”