ф-n-absorbing ideals and ф-generalized-n-absorbing ideals of commutative semirings

Abstract

In this dissertation, we introduce the concepts of ф-primary ideals, ф-n-absorbing ideals and ф-generalized-n-absorbing ideals of a commutative semiring R with nonzero identity where ф is a function from the set of ideals of R into the set of ideals of R or the empty set. These are extended from primary ideals, n-absorbing ideals and generalized n-absorbing ideals of commutative rings with nonzero identity, respectively. We investigate ф-primary ideals, ф-n-absorbing ideals and ф-generalized-n-absorbing ideals in three types of semirings structures; namely, semirings (in general), quotient semirings and semirings of fractions in various points of view. In addition, we examine relationships among ф-primary ideals, ф-n-absorbing ideals and ф-generalized-n-absorbing ideals and can conclude that ф-primary ideals and ф-n-absorbing ideals do not imply each other; nevertheless, both imply ф-generalized-n-absorbing ideals.