ideal,ring,flat module,modules over R Is there a characterization of modules (AND equivalent characterizations of rings R) over integral domains R with the property that each left ideal in R is flat?When all left ideals are projective,then we have Dedekind domains,right?
 A: You're sending some mixed signals here by referring to things which are often considered commutative (integral domains and Dedekind domains) and also referring to left modules. I'll do my best to address everything.

When all left ideals are projective,then we have Dedekind domains, right?

It's true that a commutative domain is Dedekind if and only if all of its ideals are projective, and that a commutative domain is Prufer if and only if all ideals are flat.
For noncommutative rings, a ring in which all left ideals are projective is called a left hereditary ring. If it's a domain it's just called a left hereditary domain. This turns out to be equivalent to the category of left $R$ modules being "hereditarily projective" in the sense that submodules of projective modules are projective.
Some people have found additional conditions to add in order to justify noncommutative definitions of Dedekind domains, but I don't know them well enough to comment. 

Is there a characterization of modules (AND equivalent characterizations of rings R) over integral domains R with the property that each left ideal in R is flat?

Among noncommutative rings, I don't think I've run across a special name for rings with all left ideals flat, but it's well known that it is equivalent to the category of left modules for the ring being "hereditarily flat," by which I mean that submodules of flat left modules are flat. Again, as mentioned above, this condition on a commutative domain is the same thing as a Prufer domain.

A weaker notion of "left hereditary" is "left semihereditary" where we only require the finitely generated left ideals of the ring to be flat. Among commutative domains, this is equivalent to being Prufer, but for noncommutative rings I haven't seen anything that leads me to believe that "all left ideals flat" is the same as "left semihereditary." I think left semihereditary rings have the property that left ideals are flat, though.
