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By Garrett Birkhoff

This vintage, written by way of younger teachers who grew to become giants of their box, has formed the knowledge of contemporary algebra for generations of mathematicians and continues to be a precious reference and textual content for self examine and school classes.

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Dimension of a ring and that in the next two results. R be a ring~ M be an R[x] . dimR[ x ~ and R modules generated~ it is captured let of R be a ring sequence Let of rings. between the homological ([V8]). ideal then Jensen, P and K countably ring over THEOREM 1 . 3 . 1 6 a maximal a maximal certain LEMMA 1 . 3 . 1 5 ( [ J 4 ] ) . dimR~m. dimRM. 17 ([V8]). Let R be a ring and over R~ then w . dim R. T. GLOBAL AND WEAK D I M E N S I O N S - - C I A S S I C A L SECTION 4 . RESULTS In this global section and weak d i m e n s i o n otherwise stated, discussion Atiyah, MacDonald [R2] rings; Atiyah, absolutely Boolean results rings; results to one.

Example p r e s e n t e d h e r e i s EXAMPLE. We can The due t o H a r r i s ([H2]) and Nagata ( [ N 2 ] ) . There e x i s t s a r i n g T which i s n o t c o h e r e n t and f o r which Tm i s N o e t h e r i a n f o r e v e r y maximal i d e a l m o f T. Construction of T: rings containing there is a surjective Let { R l } be a family a common map: field K. of N o e t h e r i a n Let A be an infinite @ :A--+ { RI}, local set and let B be another for which infinite set. Set C = A x B and denote by A U C the disjoint union of A and C.

For every then: = IR. ] and in xl,x2~ in S and Xlr T are x2~ . . be . . over R . . over R. coherent rings. and R a where a ~ ~, rings then and, many examples Noetherian To see (For limit x n] provides coherent. for . , polynomial R a module R , of R, let polynomial the flat and . of Ra module R a module of and of J as an R m o d u l e . ) a presentation satisfying Section R is of Xn]]. direct obtained flat J a of R a s a t i s f y i n g R be the a ideal ® RaRe) Xl, , is generated Noetherian~ R[[Xl, R system R~ i s then presentation ring S = I~[R Although is be directed % generated Set the R is Proof: ring that a finite Let over T = R[[Xl,X2~ which see ideal a finite a a ~ ~t every finitely generated w i t h R, we o b t a i n T = lin_~ for for if J is a f i n i t e l y R is R a flat, Assume that e S be ® R alii+m R~ = lim__+ ( I a finitely and~ { R ~a Suppose a is Fix and Let of hence, in such R a are R[x] rings, this, ring example is, nevertheless, without the "flatness" not R be one variable a ring see Ra[x] coherent.

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