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A ring R is said to be n-fine if every nonzero element in R can be written as a sum of a nilpotent and n units in R. The class of these rings contains fine rings and n-good rings in which each element is a sum of n units. Fundamental properties of such rings are obtained. One of the main results of this paper is that the m × m matrix ring Mm(R) over any arbitrary ring R is 2-fine. Furthermore, the m × m matrix ring Mm(R) over a n-fine ring R is n-fine. [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 be an extension of integral domains, and let be the integral closure of in . The main purpose of this paper is to study , the set of intermediate rings between and . As a main tool, we establish an explicit description of any intermediate ring in terms of localizations of (or ). This study effectively enables us to characterize the minimal extensions in . We also prove a special chain theorem concerning the length of an arbitrary maximal chain in . [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]