We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and Saorín. We also show that if a class of modules contains the ground ring and is closed under extensions, direct sums, pure submodules, and pure quotients, then it for...

Adámek and Sousa recently solved the problem of characterizing the subcategories K of a locally λ-presentable category C which are λ-orthogonal in C, using their concept of Kλ-pure morphism. We strenghten the latter definition, in order to obtain a characterization of the classes defined by orthogonality with respect to λ-presentable morphisms (where f :A B is called λ-presentable if it is a λ-...

Many mathematicians have sought ‘pure’ proofs of theorems. There are different takes on what a ‘pure’ proof is, though, and it’s important to be clear on their differences, because they can easily be conflated. In this paper I want to distinguish between two of them. I want to begin with a classical formulation of purity, due to Hilbert: In modern mathematics one strives to preserve the purity ...

The purpose of this study is to assess the factors affecting  membership satisfaction in Andimeshk, using the Cochran Formula  with 260 people as sample size. Significant relationship between the variables for the statistical tests like Summer's D, Kendall, and Spearman and multivariate regression analysis and path analysis were mental health of family and social levels, and eliminating or cont...