Prime Ideals in Semirings

In this paper, we prove the following theorems: 1. A nonzero ideal \(I\) of \((\mathbb{Z}^+,+,\cdot)\) is prime if and only if \(I=\langle p\rangle\) for some prime number \(p\) or \(I=\langle 2,3\rangle\). 2. Let \(R\) be a reduced semiring. Then a prime ideal \(P\) of \(R\) is minimal if and only...

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Bibliographic Details
Published inBulletin of the Malaysian Mathematical Sciences Society Vol. 34; no. 2
Main Authors Gupta, Vishnu, Chaudhari, JN
Format Journal Article
LanguageEnglish
Published Heidelberg Springer Nature B.V 01.01.2011
Subjects
Mathematics
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Summary:In this paper, we prove the following theorems: 1. A nonzero ideal \(I\) of \((\mathbb{Z}^+,+,\cdot)\) is prime if and only if \(I=\langle p\rangle\) for some prime number \(p\) or \(I=\langle 2,3\rangle\). 2. Let \(R\) be a reduced semiring. Then a prime ideal \(P\) of \(R\) is minimal if and only if \(P=A_P\) where \(A_P=\{r\in R:\exists \ a\notin P\) such that \(ra=0\}\). 2010 Mathematics Subject Classification: 16Y60.
ISSN:0126-6705
2180-4206