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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.2012-11-03
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    A counterexample is $R[x]/(x^2)$ for any domain $R$.2012-11-03
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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 $\det I = 1$.2012-11-03
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    @Rahul: Sure, that's a beautiful simple characterization!2012-11-03
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    @Rankeya: Corrected! Thanks.2012-11-03
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    @Rahul: What about something like this? _An alternating multilinear $n$-form on the column space of an $n \times n$ matrix over a field $F$ equals the determinant if and only if the matrix is an element of $SL(n, F)$._ Now the question is determining which fields make the statement true, i.e., characteristic not $2$, etc.2012-11-04

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