 By M. Petrich

ISBN-10: 038707998X

ISBN-13: 9780387079981

Read Online or Download Categories of Algebraic Systems: Vector and Projective Spaces, Semigroups, Rings and Lattices PDF

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Extra resources for Categories of Algebraic Systems: Vector and Projective Spaces, Semigroups, Rings and Lattices

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Since there B'C' V(P) = W 1 u W2, w h e r e sets, irreducible. then subset generic If V(P) one of t h e s e prime If P • X, t h e n B' a n d B = A + B', C' w h i c h sets, then P Conversely, let V(A) A : t a d A. If are C : A + C' x = a I + b = a 2 + c with x R x C b R c + A c A and this closed not A contained is be is not in T h e n A = B n C; A indeed, a l , a 2 e A a n d b e B, c e C. yields x • A since A is r a d i c a l . 43 Therefore, V(A) such P that does we get that bility of V(A). since As = V(B) V(A) {As, ~ • open sets compact contain the ideal ~ % entailing is a n o t h e r ideals of = X - V(A but not R A Hence, exists point a P • V(A) picking a contradiction with for V(A) an x • B - A , the then irreduciP n A but = P.

F-l(A) : f-l(B)] : f(B)]. = f-l[A If P e C(~1) , p n Ker f then f(P) e C(~2). Let f : R 1, T 1 ~ R2, T 2 be a final morphism. : B]. 41 1. If o I is restricted 2. If f then o 2 is restricted. is a final torsion reduction then o I is restricted if and only if o 2 is restricted. PROOF. 1. Let A • C'(o 2) then f-l(A) • C'(o 1) and Thus f[f-l(A ) : R 1] = [A : R 2] and [f-l(A) : R 1] • C(Ol). [A : R 2] • C(o 2) by the foregoing re- marks. 2. Since Ker f c Ol(R1) , every A • C'(o 1) contains Ker f and hence A • C'(o 1) then f(A) • C'(o2).

R be a left N o e t h e r i a n Suppose that prime ring, a R _ p is a T - f u n c t o r let such P that be a the 30 elements of G(P) are units in Q R - p ( R ) ' then : 1. ~P : ~ R - P 2. R satisfies the left Ore condition with respect to G(P). 3. ~ R - P is a r e s t r i c t e d quasi-prime. 4. a R _ p is a prime kernel functor if and only if the induced TR/A are symmetric for all A • C'(~). 5. The elements of C'(~ R _ p) are prime left ideals, they are the left ideals maximal with the property of being disjoint from G(P).