Determinants of r-fold symmetric multilinear forms (Q1121353)

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.