Module over a ring which satisfies Whitehead's axioms of projective geometry

Question

I asked this question(Characterization of a vector space over an associative division ring). It was pointed out that the answer was negative. So I reconsidered the problem and have come up with the present one.

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are:

G1: Every line contains at least 3 points

G2: Every two points, A and B, lie on a unique line, AB.

G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).

Now let $V$ be a vector space over a field $K$. We denote by $P(V)$ the set of one-dimensional subspaces of $V$. If $V$ is finite dimensional, this is the usual definition of a projective space over $K$. We say a two-dimensional subspace of $V$ a line of $P(V)$. Then points and lines of $P(V)$ satisfy the above axioms(see Whitehead's axioms of projective geometry and a vector space over a field)

Conversely let $M$ be a faithful left module over an associative ring $R$ with unity. Suppose $M$ satisfies the following conditions.

1) It has a composition series.

2) For every submodule $N$ of $M$, there exists a submodule $L$ such that $M = N + L$ and $N \cap L = 0$.

Then every submodule $N$ has a composition series. We call its length the dimension of $N$ and denote it by dim $N$. Let $P(M)$ be the set of one-dimensional submodules of $M$. An element of $P(M)$ is called a point. We say a two-dimensional submodule a line of $P(M)$. Then it is clear that $P(M)$ satisfies Axiom G2. It is not difficult to see that it also satisfies Axiom G3(see the answer to aforementioned question).

Now we add the following conditions on $M$.

3) $P(M)$ satisfies Axiom G1.

4) dim $M \ge 3$.

If $M$ satisfies the conditions 1), 2), 3), 4) is $R$ necessarily a division ring?

Answer 1

I haven't actually checked to see if this works but it seems that the $\mathbb{Z}$-module $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ might satisfy these conditions.

Answer 2

For a division ring $D$, let the matrix ring $M_3(D)=M=R$.

Now you can see the answer is no:

• $M$ is obviously faithful and has composition length $3$.
• It is a semisimple module since $R$ is a semisimple ring
• Any dimension $2$ module over $R$ looks like $S\oplus S$ where $S$ is (the unique) dimension $1$ module for $R$. This module has at least three distinct dimension $1$ submodules: $S\oplus \{0\}$, $\{0\}\oplus S$, and $\{(s,s)\mid s\in S\}$.