By Chazelle B., Goodman J.E., Pollack R. (eds.)
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Extra resources for Advances in Discrete and Computational Geometry
Example text
28D applies. 28G BIZARRE EXAMPLE. If L = C(X), then L « C, hence there is a subfield F of I such that F « R, and [ I : F] = 2. (Cf. ) In this way one can construct a countable chain of fields {Ln}<^=1 with each Ln « C. 30. A R T I N - S C H R E I E R T H E O R Y O F F O R M A L L Y R E A L F I E L D S 13 the field K — IJ^Li Ln is also algebraically closed with \K\ — |C|, hence K « C. 28A,B and D. 28H T H E O R E M (STEINITZ. cit). L/F. P R O O F . If X and Y are transcendence bases of K/F and L/F of the same cardinality, then A = F(X) « B = F(Y), and hence K = ~A « L = B are isomorphic algebraic closures of isomorphic fields.
7A). Another Twin: Ace and Dec So you have to have ascending chain conditions on certain (right) ideals, and maybe the descending chain condition (— dec) on certain ideals. 25). 31; Cf. 6). 4C). 5A), that is, satisfies the dec on all right ideals. FGC Rings Much of the survey is an elaboration of these themes. 3). The first question involves the notions of (almost) maximal rings, equivalently (almost) linearly compact rings in the discrete topology, and Bezout domains (sup. 4B), /i-local domains (sup.
REMARKS. IB, every simple R[x]-module has finite length as an Rmodule. Cf. 36D. Artinian Rings and the Hopkins-Levitzki Theorem The theorem of Artin [27] characterizes a semisimple ring R by (1) the descending chain condition (dec) on right ideals (= R is right Artinian), and (2) R has no nilpotent ideals / 0 (Cf. 34As) (= R is semiprime). The original proof required the further assumption of the ascending chain condition (ace) on right ideals, that is, that R is right Noetherian. However, the theorems of Hopkins [39] and Levitzki [39] obviated this: every right Artinian ring is right Noetherian.