(k) Let f: A B be a mapping of sets, A a system of subsets of A, and B a system ofsubsets of B. DefineS(A) = {S(X) C B : X e A).(8) = {r-)eA :Y e B}.Prove that f-1(B) is a ring if B is a ring.(1) Show that f(A) is not necessarily a ring if A is a ring.(m) Show that -(B) is a a-algebra if B is a o-algebra.(n) Prove thatR(s-(8) = (R(B).where R(C) is the minimal ring containing e.

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.3: The Characteristic Of A Ring

Problem 4E

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Let : A B be a mapping of sets, A a system of subsets of A, and 3 a system of subsets of B. Define S(A) = S(X) C B: X€ A). (8) = {) eA:Y e B}. %3D Prove that f-1(B) is a ring if B is a ring. Show that f(A) is not necessarily a ring if A is a ring. uehra