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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Codominant dimension of rings and modules


Author: Gary L. Eerkes
Journal: Trans. Amer. Math. Soc. 176 (1973), 125-139
MSC: Primary 16A60
MathSciNet review: 0314906
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Abstract: Expanding Nakayama's original concept of dominant dimension, Tachikawa, Müller and Kato have obtained a number of results pertaining to finite dimensional algebras and more generally, rings and their modules. The purpose of this paper is to introduce and examine a categorically dual notion, namely, codominant dimension. Special attention is given to the question of the relation between the codominant and dominant dimensions of a ring. In particular, we show that the two dimensions are equivalent for artinian rings. This follows from our main result that for a left perfect ring R the dominant dimension of each projective left R-module is greater than or equal to n if and only if the codominant dimension of each injective left R-module is greater than or equal to n. Finally, for computations, we consider generalized uniserial rings and show that the codominant dimension, or equivalently, dominant dimension, is a strict function of the ring's Kupisch sequence.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0314906-3
PII: S 0002-9947(1973)0314906-3
Keywords: Perfect rings, minimal injective cogenerator, minimal projective resolutions, injective projective modules, codominant and dominant dimensions, generalized uniserial rings, Kupisch sequences
Article copyright: © Copyright 1973 American Mathematical Society