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Assume that B is noetherian

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MATH-GA 2210.001: Homework Algebraic Number Theory 1

1. Let A ⊂ B be an inclusion of rings. Assume that B is noetherian. Is it true that A is noetherian?

2. Let A be a Dedekind ring and let I, J be nonzero ideals of A. Express, in terms of decomposition ofI and J as a product of powers of prime ideals, the decomposition of

I∩J, IJ, I +J.

Deduce that ifp1, . . . ,prare distinct prime ideals ofA, then Qr

i=1pi =∩ri=1pi.

3. Let I be a nonzero ideal of a Dedekind ringA and assume I 6=A. Show that I is generated by two elements.

4. Show that a Dedekind ring with finitely many prime ideals is principal.

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