Let A, m be an Artinian local ring (see 1.13 and 3.2). Prove that the maximal ideal m is nilpotent. More precisely, the d.c.c. gives that mk [Hint: suppose by contradiction that mk 0; let I be minimal among the ideals of A with I · m* # 0. Prove that I = (x) is principal, then apply Nakayama's lemma 2.8.2 to it. Compare mk+1 for some k > 0. Prove that mk = 0. %3D

Q: The doctor prescribes 25mg/kg divided into 3 equal doses for a todddler whom was 42 lbs. find the…

A: The weight of the child is 42 lbs The weight of the child in kilograms is 42×0.4536=19.0512≈19…

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Q: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

A: Question: Prove that an ideal I in a ring R is the whole ring if and only if 1∈I. Proof: We have to…

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Q: a Prove that I = { : a,b,c,d are even integers} is an ideal of M2(Z). d

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Q: (10) is maxi

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Q: Let S = {a+bi: a, b € Z and b is even}. Is S an ideal of Z[i]?

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Q: Let I = {(a, 0) | a e Z}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z.

A: Ideal of a ring

Q: I. Exercise 2.64.1 Show that if I is an ideal of a ring R, then 1 E I implies I = R.

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Q: Let R be a ring. For a subset X C R, let (X) denote the ideal generated by X, i.e., the smallest…

A: I have given detailed solution below thanks Explanation:Step 1: Step 2: Step 3: Step 4:

Q: 1. Prove that if S is a subring of a ring R then the following are true: (a) Os = OR (b) if 1R € S,…

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Q: explain if the following statement is true or false in detail: "The ideal (x + 1) is maximal in the…

A: "The ideal (x + 1) is maximal in the ring R[x, y]"