Abstract
We define the presented dimensions for modules and rings to measure how far away a module is from having an infinite finite presentation and develop ways to compute the projective dimension of a module with a finite presented dimension and the right global dimension of a ring. We also make a comparison of the right global dimension, the weak global dimension, and the presented dimension and divide rings into four classes according to these dimensions.
Highlights
Let R be a ring and n a nonnegative integer
We define the presented dimensions for modules and rings to measure how far away a module is from having an infinite finite presentation and develop ways to compute the projective dimension of a module with a finite presented dimension and the right global dimension of a ring
Following 1, 2, a right R-module M is called n-presented in case it has a finite n-presentation, that is, there is an exact sequence of right R-modules
Summary
Let R be a ring and n a nonnegative integer. Following 1, 2 , a right R-module M is called n-presented in case it has a finite n-presentation, that is, there is an exact sequence of right R-modules. The lambda dimension of a ring R is the infimum of the set of integers n such that every R-module having a finite n-presentation has an infinite finite. Ng 4 defined the finitely presented dimension of a module M as f.p. dim M inf{n | there exists an exact sequence Pn 1 â Pn â ¡ ¡ ¡ â P0 â M â 0 of R-modules, where each Pi is projective, and Pn 1, Pn are finitely generated}, which measures how far away a module is from being finitely presented. We define a dimension, called presented dimension, for modules and rings in this paper It measures how far away a module is from having an infinite finite presentation and how far away a ring is from being Noetherian. For other definitions and notations in this paper we refer to 5, 6
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