From: fiedorow@math.ohio-state.edu (Zbigniew Fiedorowicz)
Subject: Re: ring like structures
Date: 3 May 2000 16:15:54 GMT
Newsgroups: sci.math
Summary: [missing]

No, near-rings are only required to satisfy a one-sided
distributivity property, but not necessarily the other.
They are also required to satisfy a number of other properties.
They arose first in projective geometry as coordinate "rings"
of projective planes, and have played an important role in
finite group theory.

The existence of both left and right distributivity almost
forces commutativity.

Here is an exercise I give students whenever I teach
a course in homological algebra.

Exercise: Show that tensor products are uninteresting in
the category of (nonabelion) groups.

Here's a solution sketch: Let b: G\times H \to K be a bilinear
map between nonabelian groups. (Perhaps we should say
bimultiplicative instead of bilinear.)

1) Show that image(b) is contained in an abelian subgroup of K.
[Hint: expand b(xy,zw) in two different ways and compare the results.]

2) Show that b(commutator, anything)=1 and b(anything,commutator)=1.

3) Show that the universal bilinear map from G\times H is
G\times H \to G^{ab}\times H^{ab} \to G^{ab}\tensor H^{ab}
where G^{ab}=G/[G,G] is the abelianization of G, and similarly with
H^{ab}

If we apply this result to ring-like structures R in the sense of
the poster, we find that the "multiplication" factors through
R \times R \to R^{ab}\times R^{ab} \to R^{ab}\tensor R^{ab} \to R,
where R^{ab} is abelianization with respect to "addition" in R.

Zbigniew Fiedorowicz