$2$-nil primary ideals of commutative semirings

Thakur, M.; Sharma, T. R.

doi:10.22124/jart.2026.31757.1855

$2$-nil primary ideals of commutative semirings

Articles in Press

Document Type : Research Paper

Authors

M. Thakur ¹
T. R. Sharma ²

¹ Srinivasa Ramanujan Department of Mathematics‎, ‎Central University of Himachal Pradesh‎, ‎Shahpur‎, ‎India

² Department of Mathematics‎, ‎Himachal Pradesh University Regional Centre‎, ‎Dharamshala

10.22124/jart.2026.31757.1855

Abstract

‎In this paper‎, ‎we extend the concept of primary ideals in commutative rings by introducing a new class‎, ‎namely $2$-nil primary ideals‎, ‎in a commutative semiring $S$ with $1 \neq 0$‎. ‎We further investigate the structure of ideals by considering related classes such as $2$-nil ideals‎, ‎$2$-absorbing ideals‎, ‎and quasi-primary ideals‎. ‎A comprehensive framework has been developed to emphasize the significance of $2$-nil primary ideals in semiring theory‎, ‎focusing on their properties and interrelations‎. ‎We provide arguments and illustrative examples that demonstrate how $2$-nil primary ideals are connected to several well-established classes of ideals‎, ‎including prime ideals‎, ‎primary ideals‎, ‎$n$-ideals‎, ‎$2$-absorbing ideals‎, ‎and quasi-primary ideals‎. ‎Finally‎, ‎Theorem \ref{6} together with Corollaries \ref{q} and \ref{r} illustrates the applications of these ideals and highlights their significance within the theory of semirings‎.

Keywords

Main Subjects