How would you solve this tantalizing Halmos problem?

$1-ab$ invertible $\implies$ $1-ba$ invertible has a slick power series “proof” as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one? Geometric series. In a not necessarily commutative ring with unit (e.g., in the set of all $3 \times 3$ square matrices with real entries), if \$1 … Read more

What do epimorphisms of (commutative) rings look like?

(Background: In any category, an epimorphism is a morphism f:X→Y which is “surjective” in the following sense: for any two morphisms g,h:Y→Z, if g∘f=h∘f, then g=h. Roughly, “any two functions on Y that agree on the image of X must agree.” Even in categories where you have underlying sets, epimorphisms are not the same as … Read more

When is the tensor product of two fields a field?

Consider two extension fields K/k,L/k of a field k. A frequent question is whether the tensor product ring K⊗kL is a field. The answer is “no” and this answer is often justified by some particular case of the following result: Proposition Given a strict field extension k⊊ , the tensor product K\otimes_kK is not a … Read more

How to memorise (understand) Nakayama’s lemma and its corollaries?

Nakayama’s lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe around 10 times) in my life. But for some reason I just cannot get this lemma, i.e. I have tendency to … Read more

A Game on Noetherian Rings

A friend suggested the following combinatorial game. At any time, the state of the game is a (commutative) Noetherian ring ≠0. On a player’s turn, that player chooses a nonzero non-unit element of the ring, and replaces the ring with its quotient by the ideal generated by that element. The player to make the last … Read more