Nowadays, error control codes have become an essential technique to improve the reliability of various digital systems. A new type error control codes called m-spotty byte error control codes are applied to computer memory systems. These codes are essential to make the memory systems reliable. Here, we introduce the m-spotty Rosenbloom-Tsfasman weights and m-spotty Rosenbloom-Tsfasman weight enumerator of linear codes over F_q[u]/(u^k) with u^k=0. We also derive a MacWilliams type identity for m-spotty Rosenbloom-Tsfasman weight enumerator.