Recently, Xiang and Ouyang defined a (commutative unital) ring [Formula: see text] to be Nil[Formula: see text]-coherent if each finitely generated ideal of [Formula: see text] that is contained in Nil[Formula: see text] is a finitely presented [Formula: see text]-module. We define and study Nil[Formula: see text]-coherent modules and special Nil[Formula: see text]-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil[Formula: see text]-coherent ring and any special Nil[Formula: see text]-coherent ring is a Nil[Formula: see text]-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil[Formula: see text]-coherent ring (regardless of whether it is coherent). Several examples of Nil[Formula: see text]-coherent rings that are not special Nil[Formula: see text]-coherent rings are obtained as byproducts of our study of the transfer of the Nil[Formula: see text]-coherent and the special Nil[Formula: see text]-coherent properties to trivial ring extensions and amalgamated algebras.

Let f : A → B be a ring homomorphism, J be an ideal of B. In this paper, we investigate the finite conductor property that the amalgamation A f J might inherit from the ring A for some classes of ideals J and homomorphisms f. Our results generate original examples which enrich the current literature with new families of examples of non-coherent finite conductor rings.

UDC 512.5 In 2005, Enochs, Jenda, and López-Romos extended the notion of perfect rings to n -perfect rings such that a ring is n -perfect if every flat module has projective dimension less or equal than n .  Later, Jhilal and Mahdou defined a commutative unital ring R to be strongly n -perfect if any R -module of flat dimension less or equal than n has a  projective dimension less or equal than n .  Recently Purkait defined a ring R to be n -semiperfect if R ¯ = R / R a d ( R ) is semisimple and n -potents lift modulo R a d ( R ) .   We study  of three classes of rings, namely, n -perfect, strongly n -perfect, and n -semiperfect rings.  We investigate these notions in several ring-theoretic structures with an aim of construction of new original families of examples satisfying the  indicated properties and subject to various ring-theoretic properties.

Let [Formula: see text] be a ring homomorphism and let [Formula: see text] be an ideal of [Formula: see text]. In this paper, we investigate the transfer of the [Formula: see text]-property from a ring [Formula: see text] to his amalgamated algebra [Formula: see text]. Our aim is to give new and original families of [Formula: see text]-rings which are neither [Formula: see text]-rings ([Formula: see text]) nor [Formula: see text]-rings ([Formula: see text]), and examples of [Formula: see text]-rings which are neither [Formula: see text]-rings ([Formula: see text]) nor [Formula: see text]-rings ([Formula: see text]).

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