It is known that a ring $R$ is left Noetherian if and only if every left $R$-module has an injective (pre)cover. We show that $(1)$ if $R$ is a right $n$-coherent ring, then every right $R$-module has an $(n,d)$-injective (pre)cover; $(2)$ if $R$ is a ring such that every $(n,0)$-injective right $R$-module is $n$-pure extending, and if every right $R$-module has an $(n,0)$-injective cover, then $R$ is right $n$-coherent. As applications of these results, we give some characterizations of $(n,d)$-rings, von Neumann regular rings and semisimple rings.
Nálezový fond z oppida Stradonice obsahuje speciální perforované keramické tvary s rozdílnou funkcí. Vedle tradičně předpokládaného využití při výrobě mléčných produktů je akcentována souvislost určité varianty s procesem přípravy alkoholických nápojů. Rozsáhlá skupina tvarů s perforacemi náleží k technické keramice, určené k manipulaci se žhavým materiálem. Analogie vyčleněných variant se objevují od počátků výroby keramiky a varianty cedníků k filtraci tekutin od doby bronzové i v kovovém provedení. and The find inventory from the Stradonice oppidum includes special perforated ceramic shapes of different purpose. Apart from the traditionally supposed use for dairy products processing, relationship with the preparation of alcoholic beverages is stressed for certain variants. Large group of shapes with perforations belongs to technical ceramics, designed to handle hot materials. Analogies of the selected variants appear from the origins of pottery production, and there is also metal version of colanders for filtration of liquids since the Bronze Age.
Let $\mathcal G$ be an abstract class (closed under isomorpic copies) of left $R$-modules. In the first part of the paper some sufficient conditions under which $\mathcal G$ is a precover class are given. The next section studies the $\mathcal G$-precovers which are $\mathcal G$-covers. In the final part the results obtained are applied to the hereditary torsion theories on the category on left $R$-modules. Especially, several sufficient conditions for the existence of $\sigma $-torsionfree and $\sigma $-torsionfree $\sigma $-injective covers are presented.
Let R be a commutative Noetherian ring and let C be a semidualizing R-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to C which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every Gc-injective module G, the character module G+ is Gc-flat, then the class GIc(R) Ac(R) is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class GIc(R) Ac(R) is covering., Elham Tavasoli, Maryam Salimi., and Obsahuje bibliografii