Address:
Shaban Ghalandarzadeh, Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
Peyman Nasehpour, Department of Engineering Science, Golpayegan University of Technology, Golpayegan, Iran and Department of Engineering Science, Faculty of Engineering, University of Tehran, Tehran, Iran
Rafieh Razavi, Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
E-mail:
ghalandarzadeh@kntu.ac.ir
nasehpour@gmail.com
nasehpour@gut.ac.ir
rrazavi@mail.kntu.ac.ir
Abstract: 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.
AMSclassification: primary 16Y60; secondary 13B25, 13F25, 06D75.
Keywords: semiring, semiring polynomials, Gaussian semiring, cancellation ideals, invertible ideals.
DOI: 10.5817/AM2017-3-179