Determinants of r-fold symmetric multilinear forms
type: Item
instance of: scholarly article
title: Determinants of r-fold symmetric multilinear forms
zbMATH Open document ID: 0674.15020
DOI: 10.1016/0021-8693(89)90160-9
author: Teruo Kanzaki
author: Yutaka Watanabe
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: A generalized determinant as an extension of the ordinary determinant is proposed for n-fold symmetric multilinear forms. Let R be a commutative ring with identity 1. For an R-module E, let \(\theta\) :E\(\times E\times...\times E\to R\) be an n-fold symmetric multilinear form, where (E,\(\theta)\) is called an n-symmetric R-module. (E,\(\theta)\) is said to be regular if \(x_ 1\in E,\) \(\theta (x_ 1,x_ 2,...,x_ n)=0\) for all \(x_ 2,x_ 3,...,x_ n\in E\) implies \(x_ 1=0\). A determinant D(E,\(\theta)\) is then defined for an n-symmetric K-module (E,\(\theta)\) for a field K and a notion of k-regular. By using an n-fold matrix of degree n and its determinant it is shown that if n is even, then the determinant has properties similar to ordinary determinants of matrices, however if n is odd, then the determinant is nearly zero. Some examples are considered.
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 4103277
zbMATH Keywords: generalized determinant
zbMATH Keywords: symmetric multilinear forms
zbMATH Keywords: commutative ring
zbMATH Keywords: R-module
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90160-9
OpenAlex ID: W1972766847
cites work: Reduced forms and reduced Witt rings of higher level
cites work: A Grothendieck ring of higher degree forms
cites work: Notes on signatures on rings
description: scientific article; zbMATH DE number 4103277