A note on “New fundamental relation of hyperrings”

In the theory of hyperrings, fundamental relations make a connection between hyperrings and ordinary rings. Commutative fundamental rings and the fundamental relation α ∗ which is the smallest strongly regular relation in hyperrings were introduced by Davvaz and Vougiouklis (2007). Recently, another strongly regular relation named θ ∗ on hyperrings has been studied by Ameri and Norouzi (2013). Ameri and Norouzi proved that θ ∗ is the smallest strongly regular relation such that R / θ ∗ is a commutative ring. In this paper, we show that θ ∗ ≠ α ∗ and θ ∗ is not the smallest strongly regular relation. Moreover, we show that some results of Ameri and Norouzi do not hold.