Let R be a principal ideal domain, A a unitary left R-module, and p R a
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Let R be a principal ideal domain, A a unitary left R-module, and p ϵ R a prime ( =_irreducible). Let pA = {pa |a ϵ A} and A[P] = {a ϵ A| pa= 0}.
(a) R/(p) is a field.
(b) pA and A[p] are submodules of A.
(c) A/pA is a vector space over R/(p), with (r + (p))(a + pA) = ra + pA.
(d) A[p] is a vector space over R/(p), with (r + (p))a = ra.
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a Since R is a principal ideal domain and p is a prime p is a maximal ideal Therefore Rp is a field ...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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