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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

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
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