Let P be a prime ideal in R and let D = R − P denote the corresponding multiplicative submonoid. Prove that there exists a unique maximal ideal in D^−1R (that is generated by P in D^−1R).

Question

Let [latex]P[/latex] be a prime ideal in [latex]R[/latex] and let [latex]D=R-P[/latex] denote the corresponding multiplicative submonoid. Prove that there exists a unique maximal ideal in [latex]D^{-1} R[/latex] (that is generated by [latex]P[/latex] in [latex]D^{-1} R[/latex] ).

Accepted Answer

We know from Problem 218 that prime ideals of [latex]R[/latex] not intersecting [latex]D[/latex] are in one-to-one correspondence with the prime ideals of [latex]D^{-1} R[/latex]. In particular the correspondence [latex]\phi[/latex] is given by sending [latex]I[/latex] to the ideal [latex]\phi(I)[/latex] generated by [latex]I[/latex] in [latex]D^{-1} R[/latex]. It is obvious that [latex]\phi[/latex] preserves the inclusion relation: [latex]I \subset J[/latex] implies [latex]\phi(I) \subset \phi(J)[/latex]. Since the unique largest prime ideal not intersecting [latex]D[/latex] is [latex]P[/latex], there is a unique maximal ideal in [latex]D^{-1} R[/latex].

Question 1.219: Let P be a prime ideal in R and let D = R − P denote the cor......

Related Answered Questions

Prove that if R is a DVR, then R is a PID. Conversely, if R is a PID with a unique maximal ideal, then R is a DVR. ...

Verified Answer:

Suppose R is a Noetherian integral domain. We assume further that R is a local ring of Krull dimension 1. Prove that if M is the unique maximal ideal of R and if every ideal is a power of M, then R is a DVR. ...

Verified Answer:

Suppose R is a Noetherian integral domain. We assume further that R is a local ring of Krull dimension 1. Prove that if M is the unique maximal ideal of R and dimR/M M/M² = 1, then R is a DVR. ...

Verified Answer:

Suppose R is a Noetherian local ring. Prove that if M is the unique maximal ideal of R and dimR/M M/M² is a finite dimensional vector space over R/M. If d = dimR/M M/M², then prove that any generating set for M has at least d element ...

Verified Answer:

Verified Answer:

Let k be a positive integer. Prove that the following equalities are true for two ideals I and J from a ring R: (a) I^k ⊆ J ⇒ rad(I) ⊆ rad(J); (b) I^k ⊆ J ⊆ I ⇒ rad(J) = rad(I); (c) rad(IJ) = rad(I ∩ J) = rad(I) ∩ rad(J); (d) rad(rad(I)) = rad(I); ...

Verified Answer:

Suppose P is a maximal ideal of R, and S is ring that contains R. If there exists a finite number of elements s1, . . . , sn ∈ S that generate S as a ring over R (that is to say, the coefficients of the polynomial expressions in s1, . . . , sn are all from R), then prove that there are only ...

Verified Answer:

Let R ⊂ S be a pair of rings such that S is integral over R. If P is a prime ideal in R, then prove that there exists a prime ideal Q in S such that P = Q ∩ R. ...

Verified Answer:

Let D be a multiplicative submonoid of R. Prove that there exists one-toone correspondence between prime ideals not intersecting D in R and prime ideals of D^−1R. ...

Verified Answer:

Suppose R ⊂ S is an integral ring extension. Let D be a multiplicative submonoid of R. Prove that (a) D^−1S is integral over D^−1R; (b) If P is a prime ideal of S, then S/P is integral over R/P ∩ R. ...

Verified Answer: