  We investigate hypersurfaces M isometrically immersed in an (n+1)-dimensional
semi-Riemannian space of constant curvature, n > 3, such that the operator A^3,
where A is the shape operator of M, is a linear combination of the operators
A^2 and A and the identity operator Id. The main result states that on the set
U of all points of M at which the square of the Ricci operator of M is not a
linear combination of the Ricci operator and the identity operator, the
Riemann-Christoffel curvature tensor R of M is a linear combination of some
Kulkarni-Nomizu products formed by the metric tensor g, the Ricci tensor S and
the tensor S^2 of M, i.e., the tensor R satisfies on U some Roter type
equation. Moreover, the (0,4)-tensor R.S is on U a linear combination of some
Tachibana tensors formed by the tensors g, S and S^2. In particular, if M is a
hypersurface isometrically immersed in the (n+1)-dimensional Riemannian space
of constant curvature, n > 3, with three distinct principal curvatures and the
Ricci operator with three distinct eigenvalues then the Riemann-Christoffel
curvature tensor R of M also satisfies a Roter type equation of this kind.
