Problem 5. Suppose that R = (R, +,*) and S = (S, +,*) are rings, and suppose that f : R - S is a ring homomorphism that is onto. Part (a). If a E R...

Solve problem 5 which is on Abstract Algebra. Question is based on Quotient Rings.

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Problem 5. Suppose that R = (R, +,*) and S = (S, +,*) are rings, and suppose that f : R - S is a ringhomomorphism that is onto.Part (a). If a E R and x E a + ker(f), prove that f (x) = f (a).Part (b). If a, b E R are such that f (a) = f(b), prove that a - b E ker(f).Part (c). If a, b E R are such that f (a) = f(b), use Part (b) to prove that a + ker(f) = b + ker(f).Part (d). Define a function Of : LCker(/) - S according to the rule of (a + ker(f)) = f (a). Prove thatOr is a bijection.Part (e). Prove that Or is a ring isormorphism.