By I. Kaplansky
An algebraic prelude Continuity of automorphisms and derivations $C^*$-algebra axiomatics and easy effects Derivations of $C^*$-algebras Homogeneous $C^*$-algebras CCR-algebras $W^*$ and $AW^*$-algebras Miscellany Mappings retaining invertible components Nonassociativity Bibliography
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Additional resources for Algebraic and analytic aspects of operator algebras
3 Let R[X ] be a polynomial ring over R and let P be a nonzero prime ideal in R[X ] such that P ∩ R = (0). Let f (X ) ∈ P be a polynomial of least positive degree. Then P = f (X )R[X ] if f (X ) is a Sharma polynomial. Proof Assume P = f (X )R[X ] and put f (X ) = a0 X d + · · · + ad . Suppose there exists t ∈ (a0 ) such that tai ∈ (a0 ) for 1 ≤ i ≤ d. We will arrive at a contradiction. Let tai = bi a0 with bi ∈ R (i = 1, . , d). Then a0 (t X d + b1 X d−1 + · · · + bd ) = t f (X ) ∈ P, which implies that t X d + b1 X d−1 + · · · + bd−1 X + bd ∈ P since P is prime and P ∩ R = (0).
Then pS ∈ Spec(S) and pS ∩ R = p; Consequently p is an upper-prime ideal of R with respect to S. Moreover, S/ pS = (R/ p)[T ] holds, where T is an indeterminate. a) if p ⊇ Iα and p ⊇ α Iα , then pS p = S p . Also, the converse is valid. b) if α Iα , then p is not an upper-prime ideal of R with respect to S. 5 Upper-Prime, Upper-Primary, or Upper-Quasi-Primary Ideals 17 and Qi ∩ R ⊃ p − (iii) assume that p + Jα = R. Then p is an upper-prime ideal of R with respect to S. In particular, it follows that pS ∈ Spec(S), pS ∩ R = p and S/ pS = R/ p.
Since p ⊇ Jα , it follows that p ⊇ α Iα . b) By assumption, there exists a, b ∈ R such that α = b/a with a ∈ Iα \ p. Then aα = b ∈ α Iα ⊆ p ⊆ pS. Suppose that p is an upper-prime ideal of R with respect to S. Since a ∈ p and Iα ⊆ p, it holds that α ∈ pS. Hence S/ pS = R/ p. Since p + Jα = R, there is a maximal ideal m of R such that p + Jα ⊆ m. It is obvious that α ∈ pS ⊆ m S and so S/m S = R/m. But, since Jα ⊆ m, we have S/m S = R/m[T ] by (ii,a) (replacing p by m), which is a contradiction. Thus p is not an upper-prime ideal of R with respect to S.