Intereting Posts

Proof on Riemann's Theorem that any conditionally convergent series can be rearranged to yield a series which converges to any given sum.
Strange behavior of $\lim_{x\to0}\frac{\sin\left(x\sin\left(\frac1x\right)\right)}{x\sin\left(\frac1x\right)}$
What is “Russian-style” mathematics?
$V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{\dim(W_i)}$
$ a $ and $b$ are real numbers with $0 < b < a$. Prove that if $n$ is a positive integer, then $a^n – b^n \leq na^{n-1}(a – b)$.
For which values does the Matrix system have a unique solution, infinitely many solutions and no solution?
Taylor series for $e^z\sin(z)$
Integrating trigonometric function problem $\int \frac{3\sin x+2\cos x}{2\sin x+3\cos x}dx$
A familiar quasigroup – about independent axioms
Find a formula in terms of k for the entries of Ak, where A is the diagonalizable matrix:
General term of this interesting sequence
Evaluate $\lim_{n\to\infty} \log^a(n)/n^b$
My favorite proof of the generalized AM-GM inequality: where it came from?
When does Initial Value Problems have: no solutions, more than one solution, precisely one solution?
How to tell if some power of my integer matrix is the identity?

In this question all rings are commutative with identity.

Consider the following well-known statement:

(*) Let $R$ be a ring and $S$ a multiplicatively closed subset of $R$. Suppose $I$ is an ideal of $R$ maximal among those not meeting $S$. Then $I$ is prime.

- Finitely many prime ideals lying over $\mathfrak{p}$
- If $M$ and $N$ are graded modules, what is the graded structure on $\operatorname{Hom}(M,N)$?
- What is the kernel of $K \to K$, defined by $T \mapsto x$?
- Krull dimension of this local ring
- $R$ has a subring isomorphic to $R$.
- Structure of Finite Commutative Rings

There are two easy proofs of this:

(1) The direct method, i.e. a proof along the lines “Suppose every ideal properly containing $I$ meets $S$. Let $ab \in I$. Suppose $a, b \notin I$. Then $(I, a)$ meets $S$, i.e. $s = x + at$ for some $s \in S$, $x \in I$, $t \in R$. Similarly $s' = y + bt'$. Then $ss' = (x + at)(y + bt') = xy + xbt' + yat + abtt' \in S \cap I$, whence I meets $S$.

(2) Construct the ring $S^{-1}R$ and establish the description of all ideals of $R$. The result follows from the fact that $R/I \rightarrow S^{-1}R/S^{-1}I$ is an injection.

I find proof (2) much more appealing, because it is (a) far more informative, and (b) easier to remember [although in this case making up a new proof is easy enough].

Now on to the “meat” of my question: it seems to me that there are many more statements similar to (*), i.e. following this pattern:

(**) Let $I$ be an ideal maximal among those not satisfying property $P$. Then $I$ is prime.

Famous examples of such properties $P$ are “principal” and “finitely generated”.

**Are there proofs of these (and similar) facts which are “instructive”, in the sense of being similar to proof (2) above rather than (1)?**

- Why are ideals more important than subrings?
- Least rational prime which is composite in $\mathbb{Z}$?
- What is the fraction field of $R]$, the power series over some integral domain?
- Is every prime ideal in $\Pi_{n=1}^{\infty}{k}$ maximal?
- $X \to Y$ flat $\Rightarrow$ the image of a closed point is also a closed point?
- About the localization of a UFD
- Divisibility question for non UFD rings
- Finitely many prime ideals lying over $\mathfrak{p}$
- Stronger Nakayama's Lemma
- What conditions guarantee that all maximal ideals have the same height?

Yes, for some very interesting general viewpoints see

Lam and Reyes: Oka and Ako Ideal Families in Commutative Rings, and Anderson; Dobbs; and Zafrullah: Some applications of Zorn’s lemma in algebra.

- Why is the group of units mod 8 isomorpic to the Klein 4 group?
- Show that for $|f_n| \le g_n$ $\forall n$: $\lim_{n\to \infty} {\int_E g_n } = \int_E g \Rightarrow \lim_{n\to \infty} {\int_E f_n } = \int_E f$
- Strategies and Tips: What to do when stuck on math?
- what is the mutual information of three variables?
- Does linear ordering need the Axiom of Choice?
- Find the limit: $\lim \limits_{x \to 1} \left( {\frac{x}{{x – 1}} – \frac{1}{{\ln x}}} \right)$
- the numer of partial function between two sets in combinatoric way
- Proofing that the exponential function is continuous in every $x_{0}$
- A variational strategy for finding a family of curves?
- Formula to calculate difference between two dates
- Does there exist an integer $x$ satisfying the following congruences?
- Floor Function Bound?
- Uniform convergence of difference quotients to the derivative
- Physical reflections of prime-number distribution
- Proof: if $p$ is prime, and $0<k<p$ then $p$ divides $\binom pk$