Non-Commutative Algebraic Geometry: An Introduction by Freddy M. J. van Oystaeyen, Alain H. M. J. Verschoren

By Freddy M. J. van Oystaeyen, Alain H. M. J. Verschoren (auth.)

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I. I. algebra i t must be Artinian and even f i n i t e dimensional over its center. I. algebra, R/P has a classical ring of quotients which must therefore be isomorphic to QR_p(R)/pe. 22. Remark. I. algebra is a f u l l y bounded ring, hence RR-P and ~p coincide Icf. 11~7] [ ~ ] ) . 3 central localization. is a consequence of the f a c t that ~p is a More precisely : Let REAIgc be such that JCrad{o) 1. If PEY(RI) : The elements of C(P) map onto invertible elements of QR_p(R) under JR-P : R~QR-p(R)" 2.

Propgsition. Let K be a t-functor, I an ideal of R. The following state- ments are equivalent : a. For every JE~(K), I / I J is K-torsion. b. 0~(R) jK(I) is an ideal of QK(R). A kernel functor K is said to be geometric i f i t is a t-functor and i f ideals of R extend to ideals of QK(R). A kernel functor K is said to be symmetric i f ~(K) has a f i l t e r b a s i s consisting of ideals of R. tors then KAK' and KVK' ~(KAK') =~(K)n~(K'); s ~(K'). I f K and K' are kernel func- are defined by their respective f i l t e r s = the idempotent f i l t e r generated by ~(K) and One may v e r i f y that the set of kernel functors in R-mod equiped with ^ and v , turns into a complete Brouwerian l a t t i c e .

I f R is a subring of S then S is a f . i n i t e l ~ ~enerated extension of R i f there e x i s t x I , .... x nEzR(S ) such that { X l , . 29. generates S as an R-algebra. Theorem. I. I. ring R then the f o l l o w i n g properties hold : I. S is a Jacobson ring. 2. I f M is a maximal ideal of S then RnM is a maximal ideal of R and S/M has f i n i t e rank over R/R nM. 61 3. In case R is a H i l b e r t algebra over a field K then S i s a H i l b e r t a l g e b r a o v e r K. 29, cf. 30. Theorem.

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