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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
DOI: https://doi.org/10.1090/S0002-9947-1970-0265411-1
MathSciNet review: 0265411
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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.


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Keywords: Homological dimension, global dimension of rings, cardinal number corresponding to <!– MATH ${2^{{\aleph _0}}}$ –> <IMG WIDTH="34" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${2^{{\aleph _0}}}$">, direct products of rings, endomorphism rings
Article copyright: © Copyright 1970 American Mathematical Society