A developmental process, independent of morphogenetic (shape) change, that is required for the ureter to attain its fully functional state. The ureter is a muscular tube that transports urine from the kidney to the urinary bladder or from the Malpighian tubule to the hindgut. [ GOC:yaf GOC:mtg_kidney_jan10 http://www.ncbi.nlm.nih.gov/pubmed/17881463 GOC:bf ]