Ultraviolet laser radiation has been adopted in a wide range of applications as diverse as water purification, flexible displays, data storage, sterilization, diagnosis and bioagent detection(1-3). Success in developing semiconductor-based, compact ultraviolet laser sources, however, has been extremely limited. Here, we report that defect-free disordered AlGaN core-shell nanowire arrays, formed directly on a Si substrate, can be used to achieve highly stable, electrically pumped lasers across the entire ultraviolet AII (UV-AII) band (similar to 320-340 nm) at low temperatures. The laser threshold is in the range of tens of amps per centimetre squared, which is nearly three orders of magnitude lower than those of previously reported quantum-well lasers(4-6). This work also reports the first demonstration of electrically injected AlGaN-based ultraviolet lasers monolithically grown on a Si substrate, and offers a new avenue for achieving semiconductor lasers in the ultraviolet B (UV-B) (280-320 nm) and ultraviolet C (UV-C) (<280 nm) bands.