Finitely-cosmall Quotients

Year 2023, Volume 27, Issue 1, 226 - 234, 28.02.2023

Berke KALEBOĞAZ

https://doi.org/10.16984/saufenbilder.1158987

Abstract

In this paper, we first define the notion of finitely-cosmall quotient (singly-cosmall quotient) morphisms. Then we give a characterization of this new concept. We show that an epimorphism p:Y→U is a finitely-cosmall quotient (singly-cosmall quotient) if and only if for any right R-module Z any morphism g:Z→Y such that pg is a finitely-copartial isomorphism (singly-copartial isomorphism) from Z to Y with codomain U is a finitely (singly) split epimorphism. We also investigate the relation between pure-cosmall quotient and finitely-cosmall quotient (singly-cosmall quotient) morphisms. We prove that over a right Noetherian ring R, an epimorphism p:Y→U is a pure-cosmall quotient morphism if and only if p is a finitely-cosmall quotient (singly-cosmall quotient) morphism. Moreover, we obtain an example of right minimal morphisms by using finitely-cosmall quotient (singly-cosmall quotient) morphisms.

Keywords

Pure-cosmall quotient morphisms, finitely-cosmall quotient morphisms, right minimal morphisms

References

• [1] M. Ziegler, “Model theory of modules”, Annals of Pure Applied Logic, vol. 26, pp. 149-213, 1984.
• [2] E. Monari Martinez, “On pure-injective modules”, in Abelian Groups and Modules (Udine), CISM Courses and Lectures, vol. 287 (Springer, Vienna), pp. 383–393, 1984.
• [3] M. Cortés-Izurdiaga, P. A. Guil Asensio, B. Kaleboğaz, A. K. Srivastava, “Ziegler partial morphisms in additive exact categories” Bulletin of Mathematical Sciences, vol. 10, no.3, 2050012, 2020.
• [4] B. Kaleboğaz, “F-copartial morphisms”, accepted in Bulletin of the Malaysian Mathematical Sciences Society.
• [5] G. Azumaya, “Finite splitness and Finite Projectivity”, Journal of Algebra, vol. 106, pp. 114-134, 1987.
• [6] L. Mao, “Finitely Phantom Morphisms and Finitely Split Epimorphisms” Colloquium Mathematicum, vol. 160, pp. 71-87, 2020.
• [7] B. Kaleboğaz, D. Keskin Tütüncü, “On F-cosmall morphisms”, Communications faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol.71, no.4, pp. 968-977, 2022.
• [8] M. Auslander, I. Rieten, S. O. Smalø, “Represantation Theory of Artin Algebras”, Volume 36 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995.
• [9] D. Keskin Tütüncü, “Subrings of endomorphism rings associated with right minimal morphisms”, submitted.
• [10] M. Cortés-Izurdiaga, P. A. Guil Asensio, D. Keskin Tütüncü, A. K. Srivastava, “Endomorphism rings via minimal morphisms”, Mediterranean Journal of Mathematics, vol. 18, no. 152, pp. 1-16, 2021.
• [11] D. J. Fieldhouse, “Pure theories” Mathematische Annalen, vol. 184, pp. 1-18, 1969.
• [12] R. B. Warfield, “Purity and algebraic compactness for modules”, Pacific Journal of Mathematics, vol. 28, pp. 699–719, 1969.
• [13] P. M. Cohn, “On the free product of associative rings”, Mathematische Zeitschrift, vol. 71, pp. 380-398, 1959.
• [14] B. Stenström, Rings of Quotients: An Introduction to Methods of Ring Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 217 (Springer-Verlag,New York), 1975.
• [15] M. F. Jones, “f-Projectivity and flat epimorphisms”, Communications in Algebra, vol. 9, pp. 1603-1616, 1981.