A lattice-ordered ring R is called an OIRI-ring if each of its order ideals is a ring ideal. Generalizing earlier work of Basly and Triki, OIRI-rings are characterized as those f-rings R such that R/I is contained in an f-ring with an identity element that is a strong order unit for some nil l-ideal I of R. In particular, if P(R) denotes the set of nilpotent elements of the f-ring R, then R is an OIRI-ring if and only if R/P(R) is contained in an f-ring with an identity element that is a strong order unit.
Henriksen, M., S. Larson, F.A. Smith. "When is Every Order Ideal a Ring Ideal?" Commentationes Mathematicae Universitatis Carolinae, Volume 32 Number 3, 1991, pp. 411 - 416.