Let R be a principal ideal domain, A a unitary left R-module, and p R a

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.