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All rings are commutative with 1 ̸= 0. The purpose of this paper is to investigate the concept of weakly 2-irreducible ideals generalizing weakly irreducible ideals and strongly 2-irreducible ideals. We say that a proper ideal I of a ring R is weakly 2-irreducible provided that for each ideals J,K and L of R, J n K n L ⊆ I implies that either J n K ⊆ v I or J n L ⊆ v I or K n L ⊆ v I. A number of results concerning weakly 2-irreducible ideals are given. For instance, the relationships between the notions weakly 2-irreducible, 2-absorbing, 2-absorbing primary and 2-absorbing quasi-primary in different rings, has been given. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1 ≠ 0 and n a positive integer. The main purpose of this paper is to study the concepts of n -irreducible and strongly n -irreducible ideals which are generalizations of irreducible and strongly irreducible ideals, respectively. A proper ideal I of R is called n -irreducible (respectively, strongly n -irreducible) if for each ideals I 1 , ... , I n + 1 of R , I = I 1 ∩ ⋯ ∩ I n + 1 (respectively, I 1 ∩ ⋯ ∩ I n + 1 ⊆ I) implies that there are n of the I i 's whose intersection is I (respectively, whose intersection is in I). [ABSTRACT FROM AUTHOR]

Let $R\subset S$ be an extension of integral domains, with $R^{\ast }$ the integral closure of $R$ in $S$. We study the set of intermediate rings between $R$ and $S$. We establish several necessary and sufficient conditions for which every ring contained between $R$ and $S$ compares with $R^{\ast }$ under inclusion. This answers a key question that figured in the work of Gilmer and Heinzer [‘Intersections of quotient rings of an integral domain’, J. Math. Kyoto Univ.7 (1967), 133–150]. [ABSTRACT FROM PUBLISHER]