For the exact pair factor
R = Z_3[eta]/(eta^2 - 3 eta)
≅ {(a,b) in Z_3 × Z_3 : a ≡ b mod 3},the maximal ideal is
m = (3, eta) = 3(Z_3 × Z_3).Hence the radical powers are completely explicit:
m^r = 3^r (Z_3 × Z_3)for every r >= 1, so
m^r / m^(r+1) ≅ F_3^2.Because the residue ring R/m ≅ F_3 acts by scalars on each layer, every F_3-subspace of m^r/m^(r+1) lifts to an ideal. Therefore there are exactly 6 ideals between m^(r+1) and m^r:
m^(r+1),m^r.| layer | 1-dimensional lines in F_3^2 |
total ideals between layers |
|---|---|---|
m^1/m^2 |
[ (1,0) ], [ (0,1) ], [ (1,1) ], [ (1,-1) ] |
6 |
m^2/m^3 |
[ (1,0) ], [ (0,1) ], [ (1,1) ], [ (1,-1) ] |
6 |
m^3/m^4 |
[ (1,0) ], [ (0,1) ], [ (1,1) ], [ (1,-1) ] |
6 |
m^4/m^5 |
[ (1,0) ], [ (0,1) ], [ (1,1) ], [ (1,-1) ] |
6 |
m^5/m^6 |
[ (1,0) ], [ (0,1) ], [ (1,1) ], [ (1,-1) ] |
6 |
m^6/m^7 |
[ (1,0) ], [ (0,1) ], [ (1,1) ], [ (1,-1) ] |
6 |
A convenient set of lifts of the four lines is:
m^(r+1) + Z_3·(3^r,0),
m^(r+1) + Z_3·(0,3^r),
m^(r+1) + Z_3·(3^r,3^r) = 3^r R,
m^(r+1) + Z_3·(3^r,-3^r).So the pair-factor ideal theory is already richer than a single chain: every depth contributes a projective line of new ideals, not just one more radical power.