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

We construct the so-called right adjoint sequence of an n x n matrix over an arbitrary ring. For an integer m greater than or equal to 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.