Exact Ideal Classification in the Pair Factor

Work with the exact pair factor

R = {(a,b) in Z_3 × Z_3 : a ≡ b mod 3}
  ≅ Z_3[eta]/(eta^2 - 3 eta).

Its maximal ideal is

m = 3(Z_3 × Z_3),

so

m^r = 3^r(Z_3 × Z_3)

for every r >= 1.

Because

m^r / m^(r+1) ≅ F_3^2

and the residue field R/m ≅ F_3 acts by scalars, every one-dimensional F_3-subspace of m^r/m^(r+1) lifts to an ideal.

So, for each finite quotient

R_k = R / 3^k R,

every ideal is one of two types:

1. a chain ideal m^r, 0 <= r <= k;

2. a line ideal

   I_{r,lambda} = m^(r+1) + (Z/3^k Z)·v_{r,lambda},

   for some 1 <= r < k and lambda in P^1(F_3).

A convenient choice of generators is

v_{r,0}   = (3^r,0),
v_{r,1}   = (3^r,3^r),
v_{r,2}   = (3^r,-3^r),
v_{r,∞}   = (0,3^r).

This gives exactly 4 intermediate ideals between m^r and m^(r+1) for every 1 <= r < k.

quotient	number of chain ideals	number of line ideals	total ideals
R/3R	2	0	2
R/9R	3	4	7
R/27R	4	8	12
R/81R	5	12	17
R/243R	6	16	22
R/729R	7	20	27

So the pair factor is not just a single radical chain. At every positive depth it opens into a full projective line P^1(F_3) of distinct line ideals.