FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2009, VOLUME 15, NUMBER 1, PAGES 157-173

Special classes of $l$-rings

N. E. Shavgulidze

Abstract

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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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Last modified: December 2, 2009