By F. Kasch
ISBN-10: 0124003508
ISBN-13: 9780124003507
Read Online or Download Modules and Rings: A Translation of Moduln Und Ringe (L.M.S. Monographs,) [missing pp. 17-30] PDF
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Additional info for Modules and Rings: A Translation of Moduln Und Ringe (L.M.S. Monographs,) [missing pp. 17-30]
Sample text
Le. n = ±1". From this it follows that 7l. z is directly indecomposable. (3) Every simple module M is directly indecomposable for it has only 0 and M as submodules. (4) Every module M which has a largest proper submodule or, in the set of non-zero submodules, a smallest submodule, is directly indecomposable. The proof may be left to the reader. 5 FACTOR MODULES AND FACTOR RINGS The definition of factor modules holds as in the case of factor spaces of linear vector spaces since only properties of linearity are employed in the definition.
M is of finite length. o In particular the proof has shown how from composition series for A and M/A such a series for M can be manufactured. Example. lo 1:/61:. The factors of the first are 21:/61:=:;,1:/31:, (1:/61:)/(21:/61:) =:;,1:/21:, those of the second are 31:/61: =:;, 1:/21:, (1:/61:)/(31:/61:) =:;,1:/31:, from which the isomorphism of the two chains follows immediately. The significance of the Jordan-HOlder-Schreier Theorem for modules of finite length becomes clear from the following consideration.
3 (a) "¢:": Suppose Im(B, M),e M, then let v: M ~M Im(B, M) be the natural epimorphism. Since v,e 0 there is a cp E HomR (B, M) with vcp ,e 0, consequently we have Im(cp) ~ Im(B, M) in contradiction to the definition of Im(B, M). ": Since A ,e 0 there is an I E L with A(l) ,e O. e Ker(cp). Hence we have cpA (l) ,e 0, thus cpA ,e O. (b) "¢:": Suppose Ker(M, C),e 0, then let £:Ker(M, C)~M be the inclusion mapping. Since £,e 0 there is a cp E HomR (M, C) with cp£ ,e O. Consequently we have Ker(M, C) ~ Ker(cp) in contradiction to the 0 definition of Ker(M, C).
Modules and Rings: A Translation of Moduln Und Ringe (L.M.S. Monographs,) [missing pp. 17-30] by F. Kasch
by Edward
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