On the law of nullity

Abstract

In chapter 7 of (2) conditions were given for a ring to be embeddable in a skew field; in particular, it was shown that any semifir has a universal field of fractions, over which all full matrices can be inverted. This was generalized in two different directions, by Bergman (in a letter to one of the authors in 1971) and by Dicks and Sontag(7). Dicks and Sontag characterized those rings having a field of fractions in which all full matrices are inverted; they showed that this is equivalent to Sylvester's law of nullity, and further showed that this forces the ring to have weak global dimension not exceeding 2 and all finitely generated projective modules to be free. Bergman on the other hand investigated weakly semihereditary rings having a rank function on projective modules which takes values in the natural numbers. He showed that there was a homomorphism from any such ring to a field of fractions in which every full map between finitely generated projective modules is inverted. Weakly semihereditary rings with a rank function to the natural numbers are the analogue of semifirs and so it is natural to look for a characterization of rings with a rank function on projective modules such that all full maps between projective modules become invertible in a suitable field of fractions. We shall find that, as before, this is the case if and only if Sylvester's law of nullity holds with respect to the rank function, for maps between projective modules. Further, the ring must have weak global dimension at most two. This is the content of Sections 2 and 3.