The weak dimensions of Gaussian rings

We provide necessary and sufficient conditions for a Gaussian ring R to be semihereditary, or more generally, of w.dimR < 1. Investigating the weak global dimension of a Gaussian coherent ring R, we show that the only values w.dimR may take are 0,1 and oc; but that fP.dimR is always at most one. In particular, we conclude that a Gaussian coherent ring R is either Von Neumann regular, or semihereditary, or non-regular of w.dimR = oo.