On \({\mathbb{Z}}\)-valued additive functions on module category
type: Item
instance of: scholarly article
title: On \({\mathbb{Z}}\)-valued additive functions on module category
zbMATH Open document ID: 0588.13009
author: Takao Irite
published in: Proceedings of the Japan Academy. Series A
publication date: 1985-01-01T00:00:00Z
review text: Let R be a commutative ring with an identity and \({\mathcal C}(R)\) the category of unitary R-modules. For \((S,+)\), a set with binary operation \(+\), \(L:\quad {\mathcal C}(R)\to S\) is called an S-valued additive function over R if for any exact sequence \(0\to M'\to M\to M''\to 0\) in \({\mathcal C}(R)\), \(L(M)=L(M')+L(M'').\) The study begins with R a noetherian ring, and additive functions on the quotient of R by an ideal, which then can be extended to R[X], the polynomial ring in one variable. The author then considers the case \(S={\mathbb{Z}}\), and studies the \({\mathbb{Z}}\)-valued additive functions. The paper ends mentioning a rather nice result of S. Kondo on non-trivial additive functions.
Mathematics Subject Classification ID: 13D99
Mathematics Subject Classification ID: 18F30
zbMATH DE Number: 3943985
zbMATH Keywords: Grothendieck group of the category of modules
zbMATH Keywords: additive function
zbMATH Keywords: noetherian ring
MaRDI profile type: Publication
full work available at URL: pjaa.61.51
OpenAlex ID: W2036970020
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DOI: 10.3792/PJAA.61.51
description: scientific article; zbMATH DE number 3943985