Let R be a ring and s E R a nonzero element which is not a...

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Let R be a ring and s E R a nonzero element which is not a zero divisor such that for any r R there exists r'e R so that rs = sr'. Define the localization R[s-¹] of R at s as the set of equivalence classes on the set {(r, sk): rER,k>0} where (r, sk)~ (r', s) if rs' =r's. (a) Prove that this gives an equivalence relation. (b) Define + Rs¹x R[s] R[s] by [(r, s*)]+[(r, s)] = [(rs +r'sk, sk+t)]. Show the this gives a well-defined operation on R[s]. (c) Define Rs ¹] x R[s¹] → R[s] inductively by [(r.s) [(r,s)] = [(r, -¹)]-[(r",+¹)] and [(r. s)] [(r', s)] = [(rr".+¹)] when r's sr". Show the this gives a well-defined operation on Rs¹. Let R be a ring and s E R a nonzero element which is not a zero divisor such that for any r R there exists r'e R so that rs = sr'. Define the localization R[s-¹] of R at s as the set of equivalence classes on the set {(r, sk): rER,k>0} where (r, sk)~ (r', s) if rs' =r's. (a) Prove that this gives an equivalence relation. (b) Define + Rs¹x R[s] R[s] by [(r, s*)]+[(r, s)] = [(rs +r'sk, sk+t)]. Show the this gives a well-defined operation on R[s]. (c) Define Rs ¹] x R[s¹] → R[s] inductively by [(r.s) [(r,s)] = [(r, -¹)]-[(r",+¹)] and [(r. s)] [(r', s)] = [(rr".+¹)] when r's sr". Show the this gives a well-defined operation on Rs¹.