A ring R is (right) strongly prime (SP) if every nonzero twosided ideal contains a finite set whose right annihilator is zero. Examples are domains, prime Goldie rings and simple rings; however, this notion is asymmetric and a right but not left SP ring is exhibited. All SP rings are prime, and every prime ring may be embedded in an SP ring. SP rings are nonsingular, and a regular SP ring is simple; since faithful rings of quotients of SP rings are SP, the complete ring of quotients of an SP ring is simple. All SP rings are coefficient rings for some primitive group ring (a generalization of a result proved for domains by Formanek), and this was the initial motivation for their study. If the group ring RG is SP, then R is SP and G contains no nontrivial locally finite normal subgroups. Coincidentally, SP rings coincide with the ATF rings of Rubin, and so every SP ring has a unique maximal proper torsion theory, and (0) and R are the only torsion ideals.( 1 ^{1} ) A list of questions is appended.