The problem ``which semigroup rings are rings with identity'' was raised a long time ago. In [Semigroup Forum 46, No. 1, 27-31 (1993; Zbl 0787.16024)], in order to investigate the existence of identity of an orthodox semigroup ring, \textit{F. Li} asked: for a ring \(R\) with identity and a regular semigroup \(S\), if \(RS\) is a ring with identity, is \(R\langle E(S)\rangle\) a ring with identity? The aim of this paper is to discuss this problem for FIC-semigroups and completely regular semigroups. The author proves that for an FIC-semigroup or a completely regular semigroup \(S\), if \(RS\) is a ring with identity, then \(R\langle E(S)\rangle\) is a ring with identity. \(E(S)\) denotes the set of idempotents of \(S\).