Czechoslovak Mathematical Journal, first online, pp. 1-13
A note on uniformly $pr$-submodules
Khaled ALHAZMY, Rabia Nagehan UREGEN, El Mehdi BOUBA, Suat KOÇ
Received March 21, 2026. Published online September 8, 2026.
Abstract: Let $M$ be a nonzero unital module over a commutative ring $R$ with a nonzero identity. A proper submodule $N$ of $M$ is said to be a uniformly $pr$-submodule if there exists a fixed $n\in\mathbb{N}$ such that whenever $am\in N$ with ${\rm ann} (m)=0$ for some $a\in R$ and $m\in M$, then $a^n\in(N:M).$ Indeed, uniformly $pr$-submodules are the extension of uniformly $pr$-ideals to modules in the sense of R. N. Uregen (2019). In addition to giving fundamental properties of uniformly $pr$-submodules, we also examine the relations between uniformly $pr$-submodules and other classical submodules such as uniformly primary submodules, $r$-submodules, $sr$-submodules and $pr$-submodules. Furthermore, we characterize some important classes of rings/modules such as total quotient rings and simple modules in terms of our new concept.
Affiliations: Khaled Alhazmy, Department of Mathematics, Faculty of Sciences, King Khalid University, P. O. Box 9004, Abha, Saudi Arabia, e-mail: khazmy@kku.edu.sa; Rabia Nagehan Uregen, Erzincan Binali Yildirim University, Faculty of Education, Department of Mathematics and Science Education, Erzincan, Turkey, e-mail: rabia.uregen@erzincan.edu.tr; El Mehdi Bouba (corresponding author), Mathematics Department, Pluridisciplinary Faculty, Mohammed First University, B. P. 300, Selouane, Nador 62700, Morocco, e-mail:mehdi8bouba@hotmail.fr; Suat Koç, Marmara University, Department of Mathematics, Istanbul, Turkey, e-mail: suat.koc@marmara.edu.tr
