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Problem A. Let R be a commutative ring, and I C R an ideal. Introduce I for some n >1}. (a) Prove that I is an ideal of R. (Hint: To prove closure under addition use the binomial formula, observing that rn € I for all large enough n.) (b) Check that I C I. If R is noetherian prove the inclusion VI" m C I for m some positive integer m. (Here I with m factors.) (c) Show that I is a radical ideal (meaning I = MI) if and only if R/I is reduced (i.e., has no nonzero nilpotent¹ elements); verify that prime ideals are radical. (d) Take R = Z and let N > 1. Prove that NZ = NZ where N is the largest squarefree divisor of N. (More concretely, N is obtained from the prime factorization of N by replacing all exponents by one.)

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