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We construct the so-called right adjoint sequence of an n × n matrix over an arbitrary ring. For an integer m ≥ 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.

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