Download PDF by W. Brandal: Commutative Rings whose Finitely Generated Modules Decompose

By W. Brandal

ISBN-10: 3540095071

ISBN-13: 9783540095071

ISBN-10: 3540351817

ISBN-13: 9783540351818

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244 o f J. Dugundji [ 6 ] . 8, w i l l c ± I~NI . be o f importance f o r us in the I t is perhaps s u r p r i s i n g t h a t the p r o o f a c t u a l l y produces a I f one t r i e s t o show t h a t start considering arbitrary infinite ~N - N has a 3 - p o i n t d i r e c t l y , compact Hausdorff spaces. deavor, we probably would n o t i c e t h a t the subspace the real l i n e has 0 as a 3 - p o i n t . Similarly, if an i n f i n i t e o f t h i s sequence is a 3 - p o i n t . m e t r i c space has a 3 - p o i n t .

Lemma 4 . 6 : If R is an FGC r i n g , then R is a l o c a l l y almost maximal Bezout ring. Proof: We f i r s t ated i d e a l o f ideals. Let R . 1 RM I M = RMX © RMY and RMY = {0) is c y c l i c , If . In e i t h e r and so R case i s a Bezout r i n g . I Let is f i n i t e t o c o n s i d e r the case is an FGC r i n g , I be a f i n i t e l y direct RM is a v a l u a t i o n r i n g i m p l i e s I M = RM(X + y ) . 3 Using 2 . 1 ( 1 ) gener- for x,y E R RM is a v a l u a t i o n RMX = {0} or I = R(x + y ) , I is a Bezout r i n g .

T E P . with then yb = 0 Jb = { 0 } Thus r2t = rx= and B2 = Rb r E M' r ~ M' . In p a r t i c u l a r (B2) M = ( R / J ) M b E B2 • implies Rr ~ P (B2) M, ~ RM, ~ ( R / J ) M, , and and Then y E J Jr = {0} by 2 . 2 ( 1 ) 0 , t E J , x = rt . b = r(1,u) which i s a c o n t r a - and so r E M - M' . d. Theorem 5 . 6 : (S. Wiegand [ 3 6 ] ) zero prime i d e a l Proof: of RR - (M U M') R R Rxm is a local and with N M' . Rxm' FGC r i n g {J E specR: P' E specR . and Choose x E P - {0} is a chain. i s t h e unique m i n i m a l of R Rp Let P' = prime i d e a l contained has o n l y f i n i t e l y topological logical considerations.

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Commutative Rings whose Finitely Generated Modules Decompose by W. Brandal

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