Some remarks on strongly irreducible ideals
Abstract
A proper ideal I of a ring R is called strongly irreducible ideal (briefly, SI-ideal) whenever I contains the intersection of two ideals of R, I contains at least one of
those ideals. It is clear that any prime ideal is a strongly irreducible ideal. Therefore, these ideals can be considered generalizations of the prime ideals. From this point of view, in this paper we extend some results of prime ideals to SI-ideals. As an example, we show that the number of minimal SI-ideals in noetherian arithmetical rings is finite and in these rings every ideal contains a finite intersection of SI-ideals. Also we give a similar result of the prime avoidance lemma for SI-ideals.
those ideals. It is clear that any prime ideal is a strongly irreducible ideal. Therefore, these ideals can be considered generalizations of the prime ideals. From this point of view, in this paper we extend some results of prime ideals to SI-ideals. As an example, we show that the number of minimal SI-ideals in noetherian arithmetical rings is finite and in these rings every ideal contains a finite intersection of SI-ideals. Also we give a similar result of the prime avoidance lemma for SI-ideals.
Keywords
Arithmetical ring, duo ring, Goldie type ring, quasi regular, strongly irreducible ideal, strongly zero divisor
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