5. Let R be a commutative ring with identity and let ce R\{0}. Prove that if every module of R is free then c has a mult

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5 Let R Be A Commutative Ring With Identity And Let Ce R 0 Prove That If Every Module Of R Is Free Then C Has A Mult 1 (119.35 KiB) Viewed 30 times

5. Let R be a commutative ring with identity and let ce R\{0}. Prove that if every module of R is free then c has a multiplicative inverse in R. [8 Marks) (ii) For an abelian group A, let T(A) denote the torsion sub- group of A. (a) Let M, N be abelian groups and o: M N a group ho- momorphism. Show that °(T(M)) ST(N). [3 Marks] (b) Find the elements in T(R/Z). [3 Marks (i) Let G = Z3 be a free abelian group generated by ui, U2, and uz, and let H be the subgroup generated by 2u1 + 4u2, 2u2 – 2u3, 12u3 + uj. (b) Determine generators v1, V2, and v3 for G such that H is generated by hivi, h202, h3v3 with hi|h2|h3. [6 Marks] Describe G/H as a product (direct sum) of cyclic groups. 4 Marks (ii) Give the full list of all non-isomorphic abelian groups of order 315. Write their elementary divisors and invariant factors. 4 Marks (iii) Give an example of a non-trivial category, and an example of a proper subcategory of this category. [2 Marks