By H. H. Schaefer
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Extra info for Banach Lattices and Positive Operators
Xd ) is m-primary, then m is a minimal prime of (x1 , . . 27, dim R = ht m ≤ d. Thus dim R ≤ s(R). On the other hand, if dim R = ht m = d, then by the (above mentioned) converse of Krull’s Principal Ideal Theorem, there exist x1 , . . , xd ∈ R such that m is a minimal prime of (x1 , . . , xd ). But since m is the unique maximal ideal of R, it follows that (x1 , . . , xd ) is m-primary. This shows that s(R) ≤ d = dim R. Exercises Throughout the following exercises A denotes a ring. 1. Let A = k[X1 , X2 , .
Xd ∈ R such that m is a minimal prime of (x1 , . . , xd ). But since m is the unique maximal ideal of R, it follows that (x1 , . . , xd ) is m-primary. This shows that s(R) ≤ d = dim R. Exercises Throughout the following exercises A denotes a ring. 1. Let A = k[X1 , X2 , . . ] be the polynomial ring in infinitely many variables with coefficients in a field k. Prove that A is not noetherian. √ 2. Let q be an ideal of A and p = q. Show that if A is noetherian, then pn ⊆ q for some n ∈ N. Is this result valid if A is not noetherian?
7. dim A = dim B. 8 (Going Down Theorem). Assume that A and B are domains and A is normal. If q is a prime ideal of B such that q ∩ A = p, and p′ is a prime ideal of A such that p′ ⊆ p, then there exists a prime ideal q′ of B such that q′ ⊆ q and q′ ∩ A = p′ . 9. Assume that A and B are domains and A is normal. Then for any prime ideal q of B such that q ∩ A = p, we have ht p = ht q. 8. 3; the second and third assertions follow from the first one by passing to quotient rings and localizations respectively.
Banach Lattices and Positive Operators by H. H. Schaefer