Ould Chbih, Ahmed; Ben Maaouia, Mohamed Ben Faraj; Sanghare, Mamadou Localization of graded duo-rings and grading of the module of fractions on graded duo-rings. (Localisation des duo-anneaux gradués et graduation des modules de fractions sur des duo-anneaux gradués.) (French. English summary) Zbl 1426.16037 Afr. Math. Ann. (AFMA) 7, 73-88 (2018). Summary: In this paper rings are with unity and modules are unitary. It is known that in general the ring (resp. module) of fractions of a graded ring (resp. module) is not necessarily a graded ring (resp. module).We show that: – The ring of fractions of a graded duo-ring relatively to a multiplicatively closed subset generated by regular homogeneous elements is a graded ring.– The module of fractions of module on a graded duo-ring relatively to a multiplicatively closed subset generated by regular homogeneous elements is a graded module. It is known in general that the ring of fractions of a duo-ring relatively to a multiplicatively closed subset satisfying the left conditions of Ore of duo-ring is not a duo-ring when the ring of fractions of a graded duo-ring relatively to a multiplicatively closed subset satisfying the left conditions of Ore of ring is not a graded duo-ring and we show that: – The ring of fractions of a duo-ring relatively to a multiplicativcly closed subset generated by the regular central subset is a duo-ring.– The ring of fractions of a graded duo-ring relatively to a multiplicatively closed homogeneous central subset is a graded duo-ring. [–] If \(P\) is a prime ideal of a duo-ring \(A\) such as \(A/P\) is homogeneous and central, then the localization is local and graded duo-ring. Finitely, we show that: if \(A=\oplus_{n\in\mathbb{Z}}A_n\) is a graded duo-ring, \(P\) is a prime homogeneous ideal of \(A\) and \(S\) is the set of all homogeneous of \(A/P\), then: – \(Frac(A/P)\) is a graded if, any element of \((A/P)\setminus\overline 0\) is homogeneous.– \(Frac(A/P)\cong A_{PH}/P_{PH}\) if \(A\) is a domain and any element of \(A/P\) is homogeneous. MSC: 16W50 Graded rings and modules (associative rings and algebras) 16U20 Ore rings, multiplicative sets, Ore localization Keywords:duo-ring; graded ring; graded module; muliplicatively closed subset of duo-ring generated by regular homogeneous elements; homogeneous localization PDFBibTeX XMLCite \textit{A. Ould Chbih} et al., Afr. Math. Ann. (AFMA) 7, 73--88 (2018; Zbl 1426.16037)