commun. fac. sci. univ. ank. ser. a1 math. stat. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 1303-5991 2618-6470 Ankara University 10.31801/cfsuasmas.1061084 Mathematical Sciences Matematik On $\mathcal{F}$-cosmall morphisms https://orcid.org/0000-0002-4903-2244 Kaleboğaz Berke HACETTEPE UNIVERSITY https://orcid.org/0000-0002-3459-1924 Keskin Tütüncü Derya HACETTEPE UNIVERSITY 12 30 2022 71 4 968 977 01 21 2022 05 16 2022 Copyright © 1948, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 1948 Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics

In this paper, we first define the notion of $\mathcal{F}$-cosmall quotient for an additive exact substructure $\mathcal{F}$ of an exact structure $\mathcal{E}$ in an additive category $\mathcal{A}$. We show that every $\mathcal{F}$-cosmall quotient is right minimal in some cases. We also give the definition of $\mathcal{F}$-superfluous quotient and we relate it the approximation of modules. As an application, we investigate our results in a pure-exact substructure $\mathcal{F}$.

 $\mathcal{F}$-cosmall quotients right minimal morphisms $\mathcal{F}$-superfluous quotients Auslander, M., Rieten, I., Smalø S.O., Represantation Theory of Artin Algebras Volume 36 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995. Bühler, T., Exact categories, Expo. Math., 28(1) (2010), 1–69. Cortes-Izurdiaga, M., Guil Asensio, P.A., Keskin Tütüncü D., Srivastava, A.K., Endomorphism rings via minimal morphisms, Mediterr. J. Math., 18(152) (2021), 16 pages. https://doi.org/10.1007/s00009-021-01802-9 Cortes-Izurdiaga, M., Guil Asensio, P.A., Kaleboğaz, B., Srivastava, A.K., Ziegler partial morphisms in additive exact categories, Bull. Math. Sci., 10(3) 2050012 (2020), 37 pages. https://doi.org/10.1142/S1664360720500125 Fieldhouse, D. J., Pure theories, Math. Ann., 184 (1969), 1-18. Kaleboğaz, B., F-copartial morphisms, Accepted in Bull. Malaysian Math. Sci. Soc. Keller, B., Chain complexes and stable categories, Manuscripta Math., 67(4) (1990), 379–417. Keskin Tütüncü, D., Subrings of endomorphism rings associated with right minimal morphisms, submitted. Monari Martinez, E., On Pure-Injective Modules, Abelian Groups and Modules, Proceedings of the Udine Conference CISM Courses and Lectures No. 287, Springer-Verlag, Vienna-New York (1984), 383-393. https://doi.org/10.1007/978-3-7091-2814-5 29 Warfield, R. B., Purity and algebraic compactness for modules, Pacific J. Math., 28 (1969), 699–719. Zhu, H.Y., Ding N. Q., S-superfluous and S-essential homomorphisms, Acta Math. Sci., 29B(2) (2009), 391-401. https://doi.org/10.1016/S0252-9602(09)60038-2 Ziegler, M., Model theory of modules, Annals Pure Appl. Logic, 26 (1984), 149-213.