Let \(R\) be a ring with identity, let \(S\) be a semigroup, and let \(T\) be the subsemigroup of \(S\) generated by all idempotents of \(S\). The semigroup ring of \(S\) over \(R\) is denoted by \(R[S]\). The author is interested in the two following problems. Problem 1: when is \(R[S]\) a ring with identity? Problem 2: suppose that \(R[S]\) is a ring with identity, and \(S\) is regular; is it true that then \(R[T]\) is also a ring with identity? Problem 1 was raised by the reviewer [in Semigroup Forum 36, 1-46 (1987; Zbl 0629.20039)], Problem 2 has been proposed by \textit{F. Li} [ibid. 46, 27-31 (1993; Zbl 0787.16024)]. The most interesting result of the paper is the affirmative answer to problem 2 for \(S\) orthodox (Theorem 2.1). The paper contains several results concerning Problem 1 for orthodox semigroups. Denote by \(Z_ R\) the subring of \(R\) generated by the identity of \(R\). The author shows that Problem 1 for \(S\) orthodox may be reduced to the same problem for the semigroup ring \(Z_ R[B]\) for a specially chosen finite band \(B\) from \(S\). In view of this result, the author investigates in the last section of the paper Problem 1 for semigroup rings of finite bands over \(Z_ R\).