Unmixedness and the generalized principal ideal theorem

Recent work toward extending the theory of Cohen-Macaulayness to all commutative rings (both Noetherian and non-Noetherian) has led to the definition of weak Bourbaki unmixed rings (wB-unmixed) and weak Bourbaki height-unmixed rings (wB-ht-unmixed). In this work we study these unmixed-ness conditions on rings which satisfy the generalized principal ideal theorem (GPIT) or at least the principal ideal theorem (PIT). This is a natural extension from Noetherian rings since Noetherian rings satisfy GPIT and PIT. There are, however, many rings which satisfy GPIT and/or PIT which are not Noetherian. Among the results are the following: (1) In rings which satisfy GPIT, wB-ht-unmixed is equivalent to wB-unmixed. (2) Every unmixed domain (in either sense) satisfies PIT. (3) Locally Cohen-Macaulay rings (which are locally Noetherian and therefore satisfy GPIT) are unmixed. As a corollary to result (2) we also get that a Prufer domain R is wB-ht-unmixed if and only if dim (R) <= 1.