In this thesis, we de ne and characterize right pure ideals, left pure ideals, right weakly pure ideals and left weakly pure ideals in T-semirings. We also characterize right weakly regular T-semirings by the properties of right pure ideals. Next, we introduce purely prime, purely semiprime, purely irreducible, strongly irreducible pure and purely maximal ideals in T-semirings and examine their relations. More- over, we can nd relationships between right pure ideals and purely irreducible ideals...