Radicals and $l$-modules

We show that for any special class of $l$-modules, we can define a special class of $l$-rings. We prove that the special radical of an $l$-ring $R$ can be represented as the intersection of the $l$-annihilators of $l$-modules over $R$ belonging to the special class. The prime radical of an $l$-ring $R$ can be represented as the intersection of the $l$-annihilators of $l$-prime $l$-modules over $R$.