Projective Dimension of Ideals in von Neumann Regular Rings

Abstract

This paper is motivated by an attempt to solve an old problem of Wiegand, which asks whether the projective dimension of an ideal in a commutative von Neumann regular ring depends only on the lattice of idempotents in that ideal. We compute the projective dimension of some infinitely generated ideals in von Neumann regular rings. In previous work, this projective dimension, if computable, was either ‘obvious’ or the subscript of the aleph of a generating set. We give nontrivial examples which can have arbitrary preassigned projective dimension and arbitrarily large cardinality of a generating set. The paper then presents a function l from the class of all nonzero submodules of projective modules over a von Neumann regular ring to the class of all ordinals. This function depends only on the lattice of cyclic submodules of M. We show that l (М) = 0 ⇔ M is projective and l (M) ≥ pd (M). We conjecture that pd (M) < ∞ ⇒ pd (M) = l (M) for all M. Since l (M) is defined lattice theoretically, this would answer Wiegand’s question affirmatively. Even if our conjecture is false, l (М) seems like an interesting lattice invariant to explore.