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By no means trivial, a simple characterization of a mathematical structure is a simply-stated one-liner in the following sense:

Some general structure is (surprisingly and substantially) more structured if and only if the former satisfies some (surprisingly and superficially weak) extra assumption.

For example, here are four simple characterizations in algebra:

  1. A quasigroup is a group if and only if it is associative.
  2. A ring is an integral domain if and only if its spectrum is reduced and irreducible.
  3. A ring is a field if and only its ideals are $(0)$ and itself.
  4. A domain is a finite field if and only if it is finite.

I'm convinced that there are many beautiful simple characterizations in virtually all areas of mathematics, and I'm quite puzzled why they aren't utilized more frequently. What are some simple characterizations that you've learned in your mathematical studies?

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Dear @user02138: Your second assertion is not correct. It is has to be reduced as well. Irreducible only implies the nilradical is prime. –  Rankeya Nov 3 '12 at 6:42
    
A counterexample is $R[x]/(x^2)$ for any domain $R$. –  Rankeya Nov 3 '12 at 6:53
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This doesn't fit into your format, but after having only learned about determinants via the awful cofactor expansion and fuzzy notions of volumes of parallelepipeds, I was quite pleased to learn that the determinant is the unique alternating multilinear form on the columns of a matrix such that $\det I = 1$. –  Rahul Nov 3 '12 at 8:37
    
@Rahul: Sure, that's a beautiful simple characterization! –  user02138 Nov 3 '12 at 16:34
    
@Rankeya: Corrected! Thanks. –  user02138 Nov 3 '12 at 16:35
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3 Answers 3

A natural number $p$ is prime if and only if it divides $(p-1)! + 1$ (and is greater than 1).

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The width (supremum of sizes of antichains) of the cardinals is $1$ if and only if it is $<k$ for some finite $k$.

Namely, every two cardinals are comparable, if and only if there is a fixed, finite $k$ such that given $k$ distinct cardinals there are two comparable.

One immediate corollary is that if the axiom of choice fails, then there are antichains of every finite length.

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1) Let $R$ be a commutative ring. An ideal is $P \subset R$ is prime (resp. maximal) if and only if $R/P$ is a domain (resp. a field).

I have often seen authors take this to be the definition of prime and maximal ideals.

2) A scheme is integral if and only if it is reduced and irreducible.

3) A commutative ring $R$ is Noetherian if and only if all prime ideals are finitely generated.

4) A function of sets is injective (resp. surjective) if and only if it has a left (resp. right) inverse.

5) If $M$ is an n-dimensional smooth manifold, then then tangent bundle $TM$ is trivial if and only if there exist smooth vector fields $X_1, \dots, X_n$ such that for all $p \in M$, $X_i(p)$ form a basis for $T_pM$, the tangent space at $p$.

6) A square matrix with entries in a field is invertible if and only if it has a non-zero determinant.

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