- All my rings are commutative.
- By a total quotient ring (TQR) , I mean a ring whose every regular element is a unit.
Now let $R$ denote a ring and $I$ denote an ideal of $R$. The following fact is well-known:
Proposition. The ring $R/I$ is a field iff $I$ is maximal among proper ideals.
I'm wondering if there is a similar theorem for total quotient rings. Therefore, I ask:
Original Question. The ring $R/I$ is a TQR iff $I$ is... what?
Edit. I'm also interested in the following, related issue:
Second Question. Is there some weakening of freeness that holds for modules over a TQR?
If so, does this help us answer the previous question?