We construct the so-called right adjoint sequence of an matrix over an arbitrary ring. For an integer the right -adjoint and the right -determinant of a matrix is defined by the use of this sequence. Over -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.