Abstract:
We study a special class of lattice-ordered rings and a special radical. We prove that a special radical of an $l$-ring is equal to the intersection of the right $l$-prime $l$-ideals for each of which the following condition holds: the quotient $l$-ring by the maximal $l$-ideal contained in a given right $l$-ideal belongs to the special class. The prime radical of an $l$-ring is equal to the intersection of the right $l$-semiprime $l$-ideals. We introduce the notion of a completely $l$-prime $l$-ideal. We prove that $N_3(R)$ is equal to the intersection of the completely $l$-prime, right $l$-ideals of an $l$-ring $R$, where $N_3(R)$ is the special radical of the $l$-ring $R$ defined by the class of $l$-rings without positive divisors of zero.
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