New determinants and the Cayley-Hamilton theorem for matrices over Lie nilpotent rings

We construct the so-called right adjoint sequence of an $n\times n$ matrix over an arbitrary ring. For an integer $m\geq 1$ the right $m$-adjoint and the right $m$-determinant of a matrix is defined by the use of this sequence. Over $m$-Lie nilpotent rings a considerable part of the classical determinant theory, including the Cayley-Hamilton theorem, can be reformulated for our right adjoints and determinants. The new theory is then applied to derive the PI of algebraicity for matrices over the Grassmann algebra.