(i) D~(L). a = WS. a ~ ~(L). 2, aE~(L) Conversely, By Theorems a = s c m s. 4, there exists a g s, hence s ~ S such a = s. Thus a ~ ~(L). 7. THE CASES THAT (L k = L) Recall FOR c IS MINIMAL L E 2 r.

Then the one-step projectivity from [0, z + v] to [0, z + t], determined by x, provides an isomorphism between this ring and (D, +, z, x, t). Proof. For a in D w e have a" = (y + z)(x + a) in D', and a'+'b" = (y + z)[x + [w + a'][y + (w + z)(x + b')]] = (y + z)Ix + [w + a'][y + b]] --(y + z)Ix + a+b] =(a+b)'. For multiplication, the triangles,