Question

Do A and used C as question say

A. (This problem gives an explanation for the isomorphism R 1m(A) R"/1m(A'), where A, Q-IAP, with Q and P invertible.) Let R be a ring and let M, N, U, V be R-modules such that there existR module homomorphisms α : M N, β : u--w, γ: M-+ U and δ: N V such that the following diagram is commutative: (recall that commutativity of the diagram means that δ ο α γ) If γ and δ are isomorphisms, prove Coker(a) Coker(A). Hint: This result should be a corollary of problem C in Homework 6. DefineIm(a) and if you can verify the hypotheses of that problem. Im(B) and see C. Let R be a ring and let M, N be two isomorphic R-modules. Let P C M be a submodule of M and Q 드 N be a submodule of N such that P Let f: M → N and g: P → Q be R-module isomorphisms such that g(p) = f(p) for all p E P. Prove that the following hold: (1) If mM, then f(m) E if and only if mE P (2) M/P N/Q (Hint: Construct first a homomorphism M → N/Q-its definition will probably depend on f. Show it is onto. Then use the First isomorphism theorem) Remark: In the problem above: If MN and PQ. t is not guaranteed that we can construct iso- morphisms f.g as above, satisfying g(p) (p) for all pe P. If this condition is not satisfied, then the conlcusion M// N/Q in part (2) may not hold. Let's try to give an example. To make it easy, let's assume that M-N

Answer 1

nsweri- heiveninfamaton Let R be a ring et m N,U, v be R- modules Such tat the re exist -module homomorphisms ar -then g is a module. molphisms

Such -tha as d is an isomorphisms -> Hence φ is onto Im kerp Now Same vne mem is injec hive Ima

ACN Sub module A Such tRat p R-modale isomotphi sm s Co is an isomoxphism P

eline Tt is wetl defined fis an isomophi sm deAne d is oo

me m mepj by previous st sm thoe M Ker Q