$A = \bigcap_{\mathfrak{p} \in \text{Spec(A)}} A_{\mathfrak{p}} = \bigcap_{\mathfrak{m} \in \text{MaxSpec(A)}} A_{\mathfrak{m}}$

abstract-algebra ring-theory commutative-algebra

Let $x$ be contained in the intersection and consider the ideal $I:=\{a \in A : ax \in A\}$. If $\mathfrak{m}$ is any maximal ideal, then there is some $b \in A \setminus \mathfrak{m}$ such that $bx \in A$ (since $x \in A_\mathfrak{m}$). This shows $I \not\subseteq \mathfrak{m}$. Hence, $I=A$, i.e. $x \in A$.


Key Idea the set of maximal ideals contain enough witnesses to show that every proper fraction has nontrivial denominator (ideal), $ $ i.e. if a fraction $\,f\not\in A\,$ then its denominator ideal $\, {\cal D}_f = \{ d\in A\ :\ d\:\!f\in A\} \ne (1),\:$ so $\, \cal D_f\,$ is contained in some maximal ideal $\,M,\,$ hence $\,f\not\in A_M.\:$ Therefore we conclude that $ \,f\not\in \bigcap A_{M}$ over all ${M \in \text{MaxSpec(A)}}.$

Related

Inside an equilateral triangle $ABC$,an arbitrary point $P$ is taken from which the perpendiculars $PD,PE$ and $PF$ are dropped onto the sides...
Prove that $n \ln(n) - n \le \ln(n!)$ without Stirling
Projecting a Non Negative Vector onto the Simplex
Gram-Schmidt in Hilbert space?
Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$
The translations of "unless" and "except" into symbolic logic.
Irreducible Dual Representation
Proof that sum of first $n$ cubes is always a perfect square [duplicate]
Summation of Fibonacci numbers $F_n$ with $n$ odd vs. even
Proof of the identity $\int_0^{+\infty}\frac{\sin(x)}{x^\alpha}dx=\frac{\Gamma(\alpha/2)\Gamma(1-\alpha/2)}{2\Gamma(\alpha)}$ for $\alpha\in (0,2)$.
A simple curve of positive area
Artinian affine $K$-algebra [duplicate]