In this letter, the performance of non-orthogonal multiple access (NOMA) with full-duplex relaying is investigated in multi-user single-input single-output downlink systems. Contrary to the original NOMA, in the proposed NOMA scheme, the base station (BS) and two served receivers within a resource block form a relaying broadcast channel (RBC), where one of the two receivers serves as a relay as well as a receiver. To analyze the performance of the proposed scheme, a simple achievable rate region of an RBC with compress-and-forward (CF) relaying is newly derived based on the noisy network coding (NNC). Furthermore, based on this, an achievable rate region of an RBC with NNC relaying when dirty-paper coding (DPC) is applied at the BS is derived. Numerical results show that the NOMA equipped with the proposed combined coding scheme comprised of DPC and NNC relaying yields significant gain over the originally proposed NOMA