Abstract
One of the aims of this paper is to provide a short survey on the -graded, the symmetric and the left (right) generalizations of the classical determinant theory for square matrices with entries in an arbitrary (possibly non-commutative) ring. This will put us in a position to give a motivation for our main results. We use the preadjoint matrix to exhibit a general trace expression for the symmetric determinant. The symmetric version of the classical Newton trace formula is also presented in the case.
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