This work demonstrates the synaptic properties of the alloy-type resistive random-access memory (RRAM). We fabricated the HfAlOx-based RRAM for a synaptic device in a neuromorphic system. The deposition of the HfAlOx film on the silicon substrate was verified by X-ray photoelectron spectroscopy (XPS) analysis. It was found that both abrupt and gradual resistive switching could be implemented, depending on the reset stop voltage. In the reset process, the current gradually decreased at weak voltage, and at strong voltage, it tended to decrease rapidly by Joule heating. The type of switching determined by the first reset process was subsequently demonstrated to be stable switching by successive set and reset processes. A gradual switching type has a much smaller on/off window than abrupt switching. In addition, retention maintained stability up to 2000 s in both switching cases. Next, the multiple current states were tested in the gradual switching case by identical pulses. Finally, we demonstrated the potentiation and depression of the Cu/HfAlOx/Si device as a synapse in an artificial neural network and confirmed that gradual resistive switching was suitable for artificial synapses, using neuromorphic system simulation.