Achieving ferroelectricity in III-nitride (III-N) semiconductors by alloying with rare-earth elements, e.g., scandium, has presented a pivotal step toward next-generation electronic, acoustic, photonic, and quantum devices and systems. To date, however, the conventional growth of single-crystalline nitride semiconductors often requires the use of sapphire, Si, or SiC substrate, which has prevented their integration with the workhorse complementary metal oxide semiconductor (CMOS) technology. Herein, we demonstrate single-crystalline ferroelectric nitride semiconductors grown on CMOS compatible metal-molybdenum. Significantly, we find that a unique epitaxial relationship between wurtzite and body-centered cubic crystal structure can be well maintained, enabling the realization of single-crystalline wurtzite ferroelectric nitride semiconductors on polycrystalline molybdenum that was not previously possible. Robust and wake-up-free ferroelectricity has been measured, for the first time, in the epitaxially grown ScAlN directly on metal. We further propose and demonstrate a ferroelectric GaN/ScAlN heterostructure for synaptic memristor, which shows the capability of emulating the spike-time-dependent plasticity in a biological synapse. This work provides a viable path for the integration of III-N architectures with the mature CMOS technology and sheds light on the promising applications of ferroelectric nitride memristors in neuromorphic computing.