Invertible Ideals and Gaussian Semirings

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.