Codominant dimension of rings and modules

Eerkes, Gary L.

doi:10.1090/S0002-9947-1973-0314906-3

Codominant dimension of rings and modules
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by Gary L. Eerkes PDF

Trans. Amer. Math. Soc. 176 (1973), 125-139 Request permission

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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