Let A and a e A satisfy the hypotheses of Lemma 6 8 (a) Every R submodule of A is an R (p n ) module with (r (p n )a ra Conversely, every R (p n ) submodule of A is an R submodule by pullback along R R (p n ) (b) The submodule Ra is isomorphic to R (p n ) (c) The only proper ideals of the ring R (p...

a This statement relates the Rsubmodules and Rpnmodules of A It says that every Rsubmodule of A can ...

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).

(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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Algebra Graduate Texts In Mathematics 73

ISBN: 9780387905181

8th Edition

Authors: Thomas W. Hungerford

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Let A and a e A satisfy the hypotheses of Lemma 6.8. (a) Every R-submodule of A

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Algebra Graduate Texts In Mathematics 73

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