$\mathfrak p$ be a prime ideal of a commutative ring $R$ , then $R_\mathfrak p/\mathfrak pR_\mathfrak p$ is the field of fractions of $R/\mathfrak p$?

There's an obvious map $f\colon R_P\to \text{Frac}(R/P)$, given by $a/b\mapsto (a+P)/(b+P)$. You need to check:

  • If $a/b\in R_P$, then $f(a/b)\in \text{Frac}(R/P)$, i.e., $b+P\neq 0$ in $R/P$.
  • If $a/b = c/d$ in $R_P$, then $f(a/b) = f(c/d)$.
  • $f$ is a homomorphism.

To show that $f$ induces an isomorphism $R_P/PR_P\cong \text{Frac}(R/P)$, you just need to show that the kernel of $f$ is $PR_P$. Well, suppose $a/b$ is in the kernel. Then $f(a/b) = (a+P)/(b+P)$ is equal to $0$ in $\text{Frac}(R/P)$, so $a+P$ is equal to $0$ in $R/P$. So $a\in P$, and $a/b = a(1/b) \in PR_P$. Conversely, if $a/b\in PR_P$, then it is equivalent to $p/b'$ for some $p\in P$ and $b'\notin P$. And $f(p/b') = (p+P)/(b'+P) = 0$, so $a/b$ is in the kernel (you've already checked that $f$ is well-defined, so it suffices to look at just one representative of its equivalence class).

There's a more category-theoretic way to do this exercise, where you check that $R_P/PR_P$ and $\text{Frac}(R/P)$ both satisfy the universal property that a ring homomorphism from this ring to $S$ is uniquely determined by a ring homomorphism $R\to S$ which maps every element of $P$ to $0$ and every element of $R\backslash P$ to a unit, and hence they must be isomorphic (by Yoneda's Lemma). But it sounds like you might benefit more at this stage from working through the details of the more "concrete" proof.


Since localization preserves exact sequences we have that the exact sequence
$$ 0 \to \mathfrak p \to R \to R/\mathfrak p \to 0$$ becomes
$$0 \to \mathfrak p R_{\mathfrak p} \to R_{\mathfrak p} \to (R/\mathfrak p)_{\mathfrak p} \to 0$$ Hence $(R/\mathfrak p)_{\mathfrak p}=R_{\mathfrak p}/\mathfrak pR_\mathfrak p$. From this you can conclude that it is the field of fractions since $R/\mathfrak p$ is a domain and localizing by $\mathfrak p$ means inverting all nonzero elements (there are no nonzero divisor in an integral domain).

Tags:

Localization

Abstract Algebra

Commutative Algebra

Ring Theory

Maximal And Prime Ideals

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