I have a few questions on these topics. 1) There's a theorem which says that the cancellation law holds in a ring iff the ring has 0 divisors. Does this mean that there are examples of rings where the cancellation law holds, but don't have multiplicative inverses? i.e. cancellation is more general than multiplying both sides of ab = ac by a^-1, which is how I thought of it in group theory. 2) If D is an integral domain, and it's field of quotients is Quot(D). Then D is a subring of Quot(D), however I don't see how this is true. The elements of Quot(D) are equivalence classes, so I don't see how any element of D can be in Quot(D). I know that every a in D, corresponds to a/1 in Quot(D) by an isomorphism, but how does this imply D is a subset of Quot(D), when the elements are still different (D isn't made up of equivalence classes)? 3) Can someone explain heuristically why Quot(D) is the smallest field containing D? 4) Since Quot(D) is a field, so multiplicative inverses exist for nonzero elements, can we take a/b in Quot(D) to mean a^-1 * b = a * b^-1? Thanks for any help.