  In this paper, we bring together the five-dimensional Saez-Ballester~(SB)
scalar-tensor theory [1] and the induced-matter-theory~(IMT) setting [2], to
obtain a modified SB theory (MSBT) in four dimensions. Specifically, by using
an intrinsic dimensional reduction procedure into the SB field equations in
five-dimensions, a MSBT is obtained onto a hypersurface orthogonal to the extra
dimension. This four-dimensional MSBT is shown to bear distinctive new features
in contrast to the usual corresponding SB theory as well as to IMT and the
Modified Brans-Dicke Theory (MBDT)~\cite{RFM14}. It should be emphasized that
the herein appealing solutions can emerge solely from the geometrical
reductional process, from presence also of extra dimension(s) and not from any
ad-hoc matter either in the bulk or on the hypersurface.
  Subsequently, we apply the herein MSBT to cosmology and consider an extended
spatially flat FLRW geometry in a five-dimensional vacuum space-time. After
obtaining the exact solutions in the bulk, we proceed to construct, by means of
the MSBT setting, the corresponding dynamic, on the four-dimensional
hypersurface. More precisely, we obtain the (SB) components of the induced
matter, including the induced scalar potential terms. We retrieve two different
classes of solutions. Concerning the first class, we show that the MSBT yields
a barotropic equation of state for the induced perfect fluid. We then
investigate vacuum, dust, radiation, stiff fluid and false vacuum cosmologies
for this scenario and contrast the results with those obtained in the standard
SB theory, IMT and BD theory. Regarding the second class solutions, we show
that the scale factor behaves similar to a de Sitter (DeS) model. However, in
our MSBT setting, this behavior is assisted by non-vanishing induced matter
instead, without any a priori cosmological constant.
