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arXiv · 1404.1901

Invertible Ideals and Gaussian Semirings

Shaban Ghalandarzadeh·Peyman Nasehpour·Rafieh Razavi
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Abstract

In the first section of this paper, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Pr\"{u}fer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Pr\"{u}fer 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 Pr\"{u}fer and Gaussian semirings are equivalent. At last we end this paper by giving a plenty of examples of proper Gaussian and Pr\"{u}fer semirings.

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Shaban Ghalandarzadeh, Peyman Nasehpour, Rafieh Razavi. 2014-04-07. Invertible Ideals and Gaussian Semirings. https://doi.org/10.5817/am2017-3-179

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