Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Homological dimension and cardinality


Author: B. L. Osofsky
Journal: Trans. Amer. Math. Soc. 151 (1970), 641-649
MSC: Primary 16.90; Secondary 04.00
MathSciNet review: 0265411
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \{ e(i)\vert i \in \mathcal{I}\} $ be an infinite set of commuting idempotents in a ring R with 1 such that

$\displaystyle \prod\limits_{\alpha = 0}^n {e({i_\alpha })\prod\limits_{\beta = n + 1}^m {(1 - e({i_\beta })) \ne 0} } $

for $ \{ {i_\alpha }\vert \leqq \alpha \leqq n\} \cap \{ {i_\beta }\vert 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.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16.90, 04.00

Retrieve articles in all journals with MSC: 16.90, 04.00


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0265411-1
PII: S 0002-9947(1970)0265411-1
Keywords: Homological dimension, global dimension of rings, cardinal number corresponding to $ {2^{{\aleph _0}}}$, direct products of rings, endomorphism rings
Article copyright: © Copyright 1970 American Mathematical Society