Let A and a e A satisfy the hypotheses of Lemma 6.8. (a) Every R-submodule of A
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Let A and a e A satisfy the hypotheses of Lemma 6.8.
(a) Every R-submodule of A is an R/(pn)-module with (r + (pn)a = ra. Conversely, every R/(pn)-submodule of A is an R-submodule by pullback along R → R/(pn).
(b) The submodule Ra is isomorphic to R/(pn).
(c) The only proper ideals of the ring R/(pn) are the ideals generated by pi + (pn) (i = 1,2, ... , n - 1 ).
(d) R/(pn) (and hence Ra) is an injective R/(pn)-module.
(e) There exists an R-submodule C of A such that A = Ra ⊕ C.
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a This statement relates the Rsubmodules and Rpnmodules of A It says that every Rsubmodule of A can ...View the full answer
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Related Book For 
Algebra Graduate Texts In Mathematics 73
ISBN: 9780387905181
8th Edition
Authors: Thomas W. Hungerford
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