The emerging nitride ferroelectrics, such as Al_1-xSc_xN promise to significantly advance our current information technology. In particular, two-terminal memristive devices are ideal candidates for artificial intelligence accelerators and in-memory computing due to their simplicity in design, non-volatility and non-destructive readout. The recent discovery of conductive domain walls in Al_1-xSc_xN is a promising enabler for such technology, offering several benefits compared to barrier height modulation- or tunneling-based devices. First, domain walls can be highly conductive and feature high read currents (required for aggressive lateral scaling and fast access times), also in non-epitaxial films without being restricted to the technologically challenging ultrathin thickness regime ((Formula presented.) 10 nm). Second, nitride ferroelectrics are fully compatible with silicon and GaN technology on which the ferroelectric domain wall memory (FeDMEM) can be integrated with logic circuitry. Third, excellent scalability and temperature resistance of ferroelectric Al_1-xSc_xN were demonstrated, enabling scaled, low-latency edge computing under extreme environmental conditions. In this study, a FeDMEM device consisting of a Pt/Al_0.72Sc_0.28N/Pt capacitor grown on Si substrates is electrically characterized in-depth, revealing unique peculiarities in the memristive response. A read current density of 350 A/m² and an ON/OFF ratio of 20 is achieved, allowing for consistent storing of up to eight levels of information.