Homological dimension and cardinality

Abstract:

Let $\{ e(i)|i \in \mathcal {I}\}$ be an infinite set of commuting idempotents in a ring R with 1 such that \[ \prod \limits _{\alpha = 0}^n {e({i_\alpha })\prod \limits _{\beta = n + 1}^m {(1 - e({i_\beta })) \ne 0} } \] for $\{ {i_\alpha }|0 \leqq \alpha \leqq n\} \cap \{ {i_\beta }|n + 1 \leqq \beta \leqq m\} = \emptyset$. Let I be the right ideal generated by these idempotents. We show that the projective dimension of I is $n < \infty$ if and only if the cardinality of $I = {\aleph _n}$. As a consequence, a countable direct product of fields has global dimension $k + 1$ if and only if ${2^{{\aleph _0}}} = {\aleph _k}$. The same is true for a full linear ring on a countable dimensional vector space over a field of cardinality at most ${2^{{\aleph _0}}}$. On the other hand, if ${2^{{\aleph _0}}} > {\aleph _\omega }$, then any right and left self-injective ring which is not semi-perfect, any ring containing an infinite direct product of subrings, any ring containing the endomorphism ring of a countable direct sum of modules, and many quotient rings of such rings must all have infinite global dimension.