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Problem C. Let R, S be commutative rings and 4 : R S a homomorphism. 1 An element r € R is nilpotent if rn = 0 for some n 1. (a) For an ideal I C S consider its inverse image in R defined as the subset = Check that is an ideal of R. Furthermore, show that there is an injective homomorphism of rings R/4-1(I) S/I and deduce that R/4-1(I) is isomorphic to a subring of S/I. (b) Give an example showing the image of an ideal of R is not necessarily an ideal of S. What if we assume 4 is surjective; is the image of an ideal an ideal? (c) Suppose I is a prime ideal. Explain why must be a prime ideal. (d) Give an example showing that I maximal does not imply - is max- imal. (Hint: IL in Q.)

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