A scheme associated to modules over commutative rings

For any module M over a commutative ring R with identity, we consider Sch ( M ) as a certain class of prime submodules modulo the equivalence relation that two such prime submodules are the same if their colon ideals are equal. We topologize Sch ( M ) and introduce a sheaf O Sch ( M ) of commutative...

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Bibliographic Details
Published in:Communications in algebra 2024-12, Vol.52 (12), p.5289-5301
Main Authors: Parsa, Mohammad Ali, Fazaeli Moghimi, Hosein
Format: Article
Language:English
Subjects:
Affine scheme
Modules
morphism of schemes
Primary: 13C99
prime submodule
Rings (mathematics)
scheme
Secondary: 14A15
sheaf
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Summary:For any module M over a commutative ring R with identity, we consider Sch ( M ) as a certain class of prime submodules modulo the equivalence relation that two such prime submodules are the same if their colon ideals are equal. We topologize Sch ( M ) and introduce a sheaf O Sch ( M ) of commutative rings on it, which makes ( Sch ( M ) , O Sch ( M ) ) into a scheme. In particular, if M is a module over a Noetherian ring R, then ( Sch ( M ) , O Sch ( M ) ) is a locally Noetherian scheme. Among others, we give sufficient conditions for Sch ( M ) to be an affine scheme.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2024.2369150