On k-prime Ideal of a Finite Commutative Ring

Source abstract

Let mathfrakB be a finite commutative ring with unity. Let I be the k-prime ideal of mathfrakB which is a generalization of prime ideal. For any fixed integer kgeq 3, an ideal I of a ring mathfrakB is said to be k-prime whenever for any set G of nonzero, distinct, and non-unit elements of mathfrakB, the product g 1 .g 2 .g 3 ...g k belongs to I implies that the product of the elements of a proper subset of G is in I. This paper investigates the algebraic structure of k-prime ideal in finite commutative ring Z n and studies their behaviour under ideal operations. Initially it is proved that every prime ideal of a ring mathfrakB is k-prime and also every non-zero ideal is k M -prime. The behaviour of k-prime ideal under sum, product and union is then examined. Also, some ring theoretic properties related to k-prime ideals are discussed with suitable examples. Furthermore, it is proved that if I is a k-prime ideal of mathfrakB, then rad(I) is a k-prime ideal of mathfrakB. Moreover the number of radical ideals in a finite commutative ring mathfrakB is generalized. The study is further extended to minimal prime ideals over a given ideal and their relationship with k-prime ideals. A generalized formula for the length of the ideal is established. Finally, the relationship among k-prime ideals, minimal primes over an ideal and length of the ideal are illustrated through suitable examples.