By MacLane S., G. Birkhoff

ISBN-10: 0828403309

ISBN-13: 9780828403306

This booklet goals to offer smooth algebra from first rules, so that it will be

accessible to undergraduates or graduates, and this via combining ordinary

materials and the wanted algebraic manipulations with the final innovations

which make clear their which means and significance.

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**Additional resources for Algebra**

**Example text**

JE-sub-bimodule of M , having finite length as a right 22-module. ,/Q)-module, whence M' is a left A-submodule of M. Consequently, M' = M , and therefore ME has finite length. It follows that the unique (up to isomorphism) simple right C-module is finitely generated over E, and therefore CE has finite length. 9. 8). generated as a right E-module} then M = V. In ann(S/QS)s> (ii) If P = ann(S/QS)s, m 1 there is a positive integer m such that rm(Q) and such that { ( J , . . , r ~ (Q)} is the set of prime ideals of i?

1 . We begin with a specialization of a result of Warfield [41, Lemma 2]. Let T be a noetherian ring, and let R be a noetherian subring of T. Let P be a prime ideal of T, let C be the Goldie quotient ring of T/P, and let cp:T —> C be the canonical ring homomorphism. -C-bimodule composition series 0 = Co C • • • C Cm = C for C, and let Qi = a n n ^ ( C 2 / C ? _ i ) for 1 < i < m. Recall t h a t the right or left annihilator of a simple bimodule is a prime ideal. It is immediate t h a t P n R C Qi for 1 < i < m.

Assume k consisting that R is an algebra over an algebraically of central (r,6)-constants. then 8 is an inner r-derivation automorphism If R is finite dimensional of R and some positive closed subfield and r-simple, power of r is an inner of R. 6), there exist orthogonal minimal central idempotents e l 7 . . , en G R such that ei + . + e n — 1 and such that r ( e ? ) = r(e2-_j_i) for i = l , . . , n , where subscripts are considered modulo n. 2ii) it follows that each Rei is a simple fc-algebra.

### Algebra by MacLane S., G. Birkhoff

by Kevin

4.4