Download PDF by F. Kasch: Modules and Rings: A Translation of Moduln Und Ringe (L.M.S.

By F. Kasch

ISBN-10: 0124003508

ISBN-13: 9780124003507

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Read 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]

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B) There exists a homomorphism {3 : B ~ A with {3a = l A • (2) For (3: B ~ C the following are equivalent: (a) f3 is a split epimorphism. (b) There exists a homomorphism 'Y: C ~ B with {3'Y = 1 0 Proof. e. let ao be the isomorphism defined by the restriction of the domain B of a to Im(a). 10 (c). (2) "(a)~(b)": Let B=Ker({3)EBBt> and let £:BI3b~beB be the inclusion mapping of B 1 into B. Further let i3I denote the restriction of {3 onto Bt, then {31 is an isomorphism (since (3 is an epimorphism and Ker({3) n B I = 0).

In the main everything remains valid for bimodules, but there is no need to pursue this in detail. Examples of homomorphisms (1) The O-homomorphism of A into B: 0: A 3aI-i>OEB. (2) The identity injection = inclusion of a submodule A '-')0 B L:A3al-i>aEB. (3) The natural (canonical) homomorphism of a module A onto the factor module A/ where C '-')0 A: c: v:A 3 a I-i>a + C EA/C. 1 41 DEFINITIONS AND SIMPLE PROPERTIES homomorphisms; for v it follows directly from the definition of the module AIC: .

Image of a = Im(a):= {a(a)la eA}. a is an injection : ~ 'rJal, a2 E A[a (al) (Le. a = a (a2) ~ al = a2] is one-one). rjection : ~ 1m a = Cod(a) (Le. a is a mapping "onto"). a is a bijection : ~ a is an injection 1\ a is a surjection In the following, if we speak of homomorphisms of modules without indicating the side then the concepts and considerations are to be regarded as holding for a one-sided module. All is exemplified only for right modules, where it is clear that everything holding for right modules holds, as appropriate, for left modules.

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Modules and Rings: A Translation of Moduln Und Ringe (L.M.S. Monographs,) [missing pp. 17-30] by F. Kasch

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