Matrix Order of the CM Generator Algebra

For the pair node

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

take the CM generator

M = B0 ⊕ R ⊕ B3.

With rows indexed by targets and columns by sources in the order (B0,R,B3), the endomorphism order is

End_R(M)
≅
[[ Z_3,      Z_3,        0      ],
 [ (3,0)Z_3, R,          (0,3)Z_3 ],
 [ 0,        Z_3,        Z_3    ]].

Here: - the upper-right and lower-left corners vanish because Hom(B0,B3)=Hom(B3,B0)=0 - the middle row records the two branch injections into the node - the middle column records the two quotient maps out of the node - the center entry is the node ring itself

So the full CM generator algebra is a noncommutative order whose center is exactly

Z(End_R(M)) ≅ R.

This is the order-theoretic form of the node: the quiver, the branch maps, and the center all sit inside one 3 × 3 matrix order.