9
votes
2answers
443 views

Origin of the Socle of a module

Where does the notation $\mbox{Soc}(M)$ (the sum of all simple submodules of a module $M$) first appear?
2
votes
2answers
157 views

local subrings of matrix ring

When is the subring (containing 1) of a matrix ring $M_n(k)$ over a field $k$ is local? I would be grateful for every reference concerning this matter, Thank you!
11
votes
3answers
2k views

Why is a ring called a “ring”?

Why is a ring called "ring" (or Zahlring in German)? There seems to (naive) me nothing more ring-like to a ring than there is to a group or a field. I am particularly interested to learn why the ...
11
votes
1answer
924 views

At what point does number theory stop playing with finite rings?

Basic results in number theory, like the Chinese remainder theorem, the Euclidean algorithm and Euler's theorem, are really about finite structures, namely the rings $\mathbb{Z}/n\mathbb{Z}$ for ...
5
votes
2answers
333 views

Hall's treatment of algebraic operations

Marshall Hall, in his famous book Theory of Groups, does not always require a binary operation be "well-defined", i.e. an operation is a relation instead of a function (there might be more than one ...
21
votes
4answers
3k views

Gossip about Grothendieck and distributive lattices

In Gian-Carlo Rota's Indiscrete Thoughts, there a list of mathematical gossip among which one reads: [...] What would have happened [...] if Grothendieck had known the theory of distributive ...
5
votes
1answer
1k views

What are hypergroups and hyperrings good for?

I came across the concept of a hyperring in two recent papers by Connes and Consani (From monoids to hyperstructures: in search of an absolute arithmetic and The hyperring of adèle classes). It's a ...
8
votes
1answer
513 views

what was Hilbert's geometric construction in his 17th problem?

Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...