### AIMS Mathematics

2022, Issue 12: 20767-20780. doi: 10.3934/math.20221138
Research article

# Regular local hyperrings and hyperdomains

• Received: 14 July 2022 Revised: 09 September 2022 Accepted: 14 September 2022 Published: 26 September 2022
• MSC : 20N20, 13E05

• This paper falls in the area of hypercompositional algebra. In particular it focuses on the class of Krasner hyperrings and it studies the regular local hyperrings. These are Krasner hyperrings $R$ with a unique maximal hyperideal $M$ having the dimension equal to the dimension of the vectorial hyperspace $\frac{M}{M^2}$. The aim of the paper is to show that any regular local hyperring is a hyperdomain. For proving this, we make use of the relationship existing between the dimension of the vectorial hyperspaces related to the hyperring $R$ and to the quotient hyperring $\overline{R} = \frac{R}{\langle a\rangle}$, where $a$ is an element in $M\setminus M^2$, and of the regularity of $\overline{R}$.

Citation: Hashem Bordbar, Sanja Jančič-Rašovič, Irina Cristea. Regular local hyperrings and hyperdomains[J]. AIMS Mathematics, 2022, 7(12): 20767-20780. doi: 10.3934/math.20221138

### Related Papers:

• This paper falls in the area of hypercompositional algebra. In particular it focuses on the class of Krasner hyperrings and it studies the regular local hyperrings. These are Krasner hyperrings $R$ with a unique maximal hyperideal $M$ having the dimension equal to the dimension of the vectorial hyperspace $\frac{M}{M^2}$. The aim of the paper is to show that any regular local hyperring is a hyperdomain. For proving this, we make use of the relationship existing between the dimension of the vectorial hyperspaces related to the hyperring $R$ and to the quotient hyperring $\overline{R} = \frac{R}{\langle a\rangle}$, where $a$ is an element in $M\setminus M^2$, and of the regularity of $\overline{R}$.

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