The Ideal of Semiring of The Non-negative Integer Aisyah Nur Adillah (a*), Fitriana Hasnani (b), Meryta Febrilian Fatimah (c), Nikken Prima Puspita (a)
a) Mathematics Department, Faculty of Sciences and Mathematics, Diponegoro University
Jl. Prof Jacub Rais, Semarang 50275, Indonesia
*aisyahnuradillah[at]students.undip.ac.id
b) Graduated Student of The Magister of Mathematics, Faculty of Mathematics and Natural Sciences, Gadjah Mada University
Yogyakarta 55281, Indonesia
c) Mathematics Study Program, Faculty of Mathematics and Natural Sciences, Sulawesi Barat University
Jl. Prof Dr. Baharuddin Lopa, S. H, Majene 91412, Indonesia
Abstract
A semiring is a generalization of a ring. Let (S,+,.) be a semiring. An ideal on a semiring defined analogue with the ideal on a ring. An ideal I of S is irreducible if I is an intersection ideal from any ideal A and B on S then I = A or I = B. We also known the strongly notion on irreducible concept. The ideal I of S is a strongly irreducible ideal when I is a subset of the intersection of A and B (ideal of S), then I is a subset of A or I is a subset of B. Here, we discussed the characteristics of the semiring of the non-negative integer set. We showed that kZ^{+} is an ideal of a semiring of the non-negative integer Z^{+} over addition and multiplication. In fact, kZ^{+} is a prime ideal and also a strongly irreducible ideal of the semiring Z^{+} if k is a prime number.
Keywords: Semiring- Non-negative integer set- prime ideal- strongly irreducible ideal
