Geodesic
Invertible ideals and Gaussian semirings
Archivum mathematicum, Tome 53 (2017) no. 3, pp. 179-192
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In the first section, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as $(I + J)(I \cap J) = IJ$ for all ideals $I$, $J$ of $S$. In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family of semirings, the concepts of Prüfer and Gaussian semirings are equivalent. At last, we end this paper by giving a plenty of examples for proper Gaussian and Prüfer semirings.
In the first section, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as $(I + J)(I \cap J) = IJ$ for all ideals $I$, $J$ of $S$. In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family of semirings, the concepts of Prüfer and Gaussian semirings are equivalent. At last, we end this paper by giving a plenty of examples for proper Gaussian and Prüfer semirings.
DOI : 10.5817/AM2017-3-179
Classification : 06D75, 13B25, 13F25, 16Y60
Keywords: semiring; semiring polynomials; Gaussian semiring; cancellation ideals; invertible ideals 
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Ghalandarzadeh, Shaban; Nasehpour, Peyman; Razavi, Rafieh. Invertible ideals and Gaussian semirings. Archivum mathematicum, Tome 53 (2017) no. 3, pp. 179-192. doi: 10.5817/AM2017-3-179

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