  Considering an arbitrary dimensional FLRW universe in the framework of a
generalized S\'{a}ez--Ballester (SB) theory, we establish a noncommutative (NC)
cosmological model. We concentrate on the predictions of NC model and compare
them with their commutative counterparts in both the classical and quantum
regimes. For the classic case, taking a very small NC parameter, we apply two
different methods to analyze the model features. First, we show through
numerical analysis that our NC model is a successful inflationary model capable
of overcoming the graceful exit and horizon problems. Furthermore, the NC
traces are visible the late time, which supports the UV/IR mixing
characteristic of the NC models. In the second method, we show that our NC
model can correspond to the previously developed NC inflationary models. In the
commutative quantum case, we obtain an exact wave function and then use the WKB
approximation to show that the solutions of the corresponding classical regime
are recovered. Finally, with regard to the NC quantum level, we focus on the
special case for which we show that a constant of motion exists. The latter
helps us to conveniently transform the corresponding complicated NC-WDW
equation into an ordinary differential equation, which can be easily solved
numerically for the general case or analytically for some special cases. The
resultant solutions show a damping behavior in the wave function associated
with the proposed NC model, which may be important in determining the viable
initial states for the very early universe.
