In this paper, we introduce the notion of an $l$-prime $l$-ideal and that of a right $l$-semiprime $l$-ideal. We prove that our definitions coincide with the definitions of M. A. Shatalova in the case of two-sided $l$-ideals. Our main results are the following ones. The radical of an $l$-ring can be represented as the intersection of the right $l$-ideals for each of which the following condition holds: the quotient ring by the maximal $l$-ideal contained in the given right $l$-ideal is semisimple. The hypernilpotent radical of an $l$-ring can be represented as the intersection of the right $l$-semiprime ideals satisfying the same condition.