BACKGROUND The Malassezia genus includes mainly lipophilic yeasts belonging to the cutaneous microbiota of man and other mammals. Some Malassezia species have been associated with various dermatological diseases. The factors permitting the transformation of yeasts of the Malassezia genus from a commensal organism to a pathogenic agent are still little known but the production of various enzymes...

in this paper‎, ‎we introduce the notion of $(m,n)$-‎algebr‎aically compact modules as an analogue of algebraically‎ ‎compact modules and then we show that $(m,n)$-algebraically‎ ‎compactness‎ ‎and $(m,n)$-pure injectivity for modules coincide‎. ‎moreover‎, ‎further characterizations of a‎ ‎$(m,n)$-pure injective module over a commutative ring are given‎.