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Keywords:
semiprime ideal; prime ideal; congruence of a lattice; allele; lattice polynomial; meet-irreducible element; kernel; forbidden exterior quotients; $D$-radical; prime radical
semiprime ideal; prime ideal; congruence of a lattice; allele; lattice polynomial; meet-irreducible element; kernel; forbidden exterior quotients; $D$-radical; prime radical
Summary:
The author studies some characteristic properties of semiprime ideals. The semiprimeness is also used to characterize distributive and modular lattices. Prime ideals are described as the meet-irreducible semiprime ideals. In relatively complemented lattices they are characterized as the maximal semiprime ideals. $D$-radicals of ideals are introduced and investigated. In particular, the prime radicals are determined by means of $\hat C$-radicals. In addition, a necessary and sufficient condition for the equality of prime radicals is obtained.
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