Summary of the Structure of FPF Rings

Summary

In Chapters 3 and 4 the structure of nonsingular one sided FPF rings is largely given. It is shown that they are nonsingular iff they are semiprime and that they are nonsingular on both sides. The maximal quotient ring is shown to be a two sided maximal quotient ring and FPF on both sides. The embedding of the ring in its maximal quotient ring is a flat epimorphism. Then von Neumann regular FPF rings are shown to be precisely the self-injective (both sides) rings of bounded index, and hence FPF on both sides. If besides nonsingular, the condition of finite Goldie dimension is imposed, then an FPF ring must be a semiprime Goldie ring on both sides. If the further restriction of A.C.C. on left and right ideals is added, then the ring is a bounded Dedekind domain and CFPF and conversely.

We do not know of a nonsingular FPF ring which is not semihereditary. If all nonsingular FPF rings are semihereditary, they are Baer rings as are all the finite matrix rings of said rings. The converse also holds, namely, if a ring is an FPF Baer ring, as well as the finite matrix rings over it, the ring is semihereditary. For the case of commutative FPF rings we do know if the nonsingular ones are hereditary (see the next section of this summary). A right Noetherian nonsingular FPF Cohen ring is hereditary.