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Localization Functor S-1() and Functors ext and tor
Daouda Faye and Mohamed Ben Faraj BEN MAAOUIA
Associated member, Directory of Algebraic Laboratory,
Cryptography, Codes and Applications (LACCA)- UFR SAT
Université Gaston Berger, Saint-Louis (UGB), Sénégal
Mamadou SANGHARÉ
Member, Directory of Algebraic Laboratory,
Cryptography, Algebraic Geometry and Applications
LACGAA - Technologies and Sciences Faculty
Université Cheikh Anta Diop, Dakar (UCAD), Sénégal
Mathematics Subject Classification: (2010)
Key words: Ring, duo-ring, left (right) Ore’s conditions, multiplicatively closed
subset, left A-module, ring of fractions, module of fractions, category A-Mod, Mod-A,
functors S-1(), Ext and Tor.
Abstract:
In this paper, unless otherwise stated, B is a duo-ring, A a subring of B, S
a multiplicatively closed subset of the ring A satisfying the left (right respectively) Ore’s
conditions, S the set of regular elements of S, AM a left A-module, BNA a
(B -A)-bimodule (respectively ANB a (A-B)-bimodule), S-1A the ring of fractions
of A on S ; particularly, if A is a duo-ring and P is a prime ideal of A then S is the set
of regular elements of A - P.
We establish the following results :
1) Isomorphisms of left (S)-1B-modules :
where M is an (A - B)-bimodule and N a left A-module ;
2) Isomorphisms of left (S)-1B-modules :
3) Isomorphisms of right (S)-1B-modules :
4) If B is noetherian, BMA a (B - A)-bimodule, then the functors :
TornS-1A (S-1M,-) : S-1A - Mod → (S)-1B - Mod are adjoint.