n-Ary k-absorbing hyperideals in krasner (m, n)-hyperrings

• Published in: Afrika mathematica 2022-03, Vol.33 (1), Article 19
• Main Authors: Yassine, A., Nikmehr, M. J., Nikandish, R.
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; History of Mathematical Sciences; Mathematics; Mathematics and Statistics; Mathematics Education; Quotients
• ISSN: 1012-9405; 2190-7668
• DOI: 10.1007/s13370-022-00961-6

Let ( R ,  f ,  g ) be a commutative Krasner ( m ,  n )-hyperring with the scalar identity 1 R and k ( < n ) be a positive integer. In this paper, the concept of n -ary k -absorbing hyperideal of R , as a generalization of n -ary prime hyperideal, is introduced and some related properties are investigated. A proper hyperideal I of R is called n -ary k -absorbing if whenever g ( x 1 n ) ∈ I for x 1 n ∈ R , then there are k of the x i ’s whose g -product is in I . It is proved that the radical of an n -ary k -absorbing hyperideal I is an n -ary k -absorbing hyperideal and g ( x ( k ) , 1 R ( n - k ) ) ∈ I for each x ∈ I ( m , n ) . Among other things, we show that n -ary k -absorbing hyperideal has at most k minimal n -ary prime hyperideals. Finally, the notion of the n -ary hyperideal quotient I x , where x ∈ R , is introduced and studied.