A very important example of almost strongly minimal theories are the algebraically closed fields. A. Macintyre has shown [3] that every ω1-categorical field is algebraically closed. Therefore every ω1-categorical field is almost strongly minimal. It will be shown that not every ω1-categorical ring is almost strongly minimal.

Let R0 be the factor ring C[y/(y2), where C[y] is the ring of polynomials in the indeterminate y over the field of complex numbers and (y2) the ideal generated by y2 in C[y].

It is straightforward to prove that R0 has the following properties:

1. R0 is a commutative ring with identity.

2. R0 is of characteristic 0.

3. For every polynomial p(x) = ∑ a1x1 ∈ R0[x] with of ai2 ≠ 0 for some i > 0 there is an a ∈ R0 such that p(a) · p(a) = 0.

4. For all x, y ∈ R0 such that x2 = 0 and y ≠ 0 there exists a z ∈ R0 with y · z = x.

5. There is an x ≠ 0 such that x2 = 0.

These properties can be ∀∃-axiomatised in a countable first order logic (see [4]). Let T be the set of these sentences. With Theorem 7 we get that T is model-complete.

If R is a model of T then I shall denote {a ∈ R ∣ a2 = 0}.