This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The “generic” case, where all the entries are distinct variables has been studied extensively (cf. [1] and [2] for a thorough account.) While some special cases, such as catale...

For M an R-module and m an element of M we define the order ideal, 0M(m), to be the ideal which consists of the images of m under homomorphisms of M to R. The proof proceeds by considering the height of particular order ideals, 0M(m), in relation to the projective dimension of M. Specifically, if M is a kth syzygy of the 7?-module TV and if xx, . . . ,xk_l is an R sequence, then Torf (TV, R/(xi...