An ideal [Formula: see text] of a ring [Formula: see text] is called pseudo-irreducible if [Formula: see text] cannot be written as an intersection of two comaximal proper ideals of [Formula: see text]. In this paper, it is shown that the maximal spectrum of [Formula: see text] is Noetherian if and only if every proper ideal of [Formula: see text] can be expressed as a finite intersection of pseudo-irreducible ideals. Using a result of Hochster, we characterize all [Formula: see text] quasi-compact Noetherian topological spaces.

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A proper ideal [Formula: see text] of a commutative ring [Formula: see text] is called lifting whenever idempotents of [Formula: see text] lift to idempotents of [Formula: see text]. In this paper, many of the basic properties of lifting ideals are studied and we prove and extend some well-known results concerning lifting ideals and lifting idempotents by a new approach. Furthermore, we give a necessary and sufficient condition for every proper ideal of a commutative ring to be a product of pairwise comaximal lifting ideals.

In this paper, the notion of completely strongly hollow ideals (respectively, strongly hollow elements) of a commutative ring as a generalization of strongly hollow ideals (respectively, local idempotents) is introduced and some related properties are investigated. Also, some characteristics of completely strongly hollow ideals and strongly hollow elements of some special classes of commutative rings are given. Then, we consider the decomposition of nonzero ideals of a ring into a product (respectively, (finite) sum) of completely strongly hollow ideals. Some other results are obtained too.