Rings with projective socles where studied by W. K. Nicholson and J. F. Watters [in Proc. Am. Math. Soc. 102, No. 3, 443-450 (1988; Zbl 0657.16015)]. The author extends this investigation to a larger class of rings; namely, rings with flat socles. He calls a right \(R\)-module \(M\) a flat socle module if the socle of \(M\) is a flat right \(R\)-module. A ring \(R\) is said to be a (right) flat socle ring if \(R_R\) is a flat socle module. The author shows that a direct product of a family of rings is a flat socle ring if and only if each factor is a flat socle ring, that \(R[x]\) is a flat socle ring when \(R\) is a flat socle ring and that if \(R\) and \(S\) are Morita equivalent rings, then \(S\) is a flat socle ring whenever \(R\) is a flat socle ring. The author also shows by example that flat socle rings are not left-right symmetric.
The author proves several conditions which are equivalent to a ring being a flat socle ring. He shows that flat socle rings are precisely those rings which admit a faithful socle module. This in turn is equivalent to the condition that for every maximal right ideal \(K\) of \(R\) either the left annihilator of \(K\) in \(R\) is zero or \(R/K\) is a flat right \(R\)- module. The obvious question to ask is, when does the class of flat socle rings coincide with the class of projective socle rings? The author gives several conditions which are sufficient for these classes to coincide. However, it seems that a set of necessary and sufficient conditions remains to be discovered.