Baer submodules of modules over commutative rings

Adam Anebrı; Hwankoo Kım; Najib Mahdou

doi:10.24330/ieja.1252741

Research Article

Baer submodules of modules over commutative rings

Year 2023, Volume: 34 Issue: 34, 31 - 47, 10.07.2023

Adam Anebrı Hwankoo Kım Najib Mahdou

https://doi.org/10.24330/ieja.1252741

Abstract

Let $R$ be a commutative ring and $M$ be an $R$-module. A submodule $N$ of $M$ is called a d-submodule $($resp., an fd-submodule$)$ if $\ann_R(m)\subseteq \ann_R(m')$ $($resp., $\ann_R(F)\subseteq \ann_R(m'))$ for some $m\in N$ $($resp., finite subset $F\subseteq N)$ and $m'\in M$ implies that $m'\in N.$ Many examples, characterizations, and properties of these submodules are given. Moreover, we use them to characterize modules satisfying Property T, reduced modules, and von Neumann regular modules.

Keywords

Baer submodule, d-submodule, fd-submodule, 0-submodule, annihilator

References

D. D. Anderson and S. Chun, Annihilator conditions on modules over commutative rings, J. Algebra Appl., 16(7) (2017), 1750143 (19 pp).
A. Anebri, N. Mahdou and A. Mimouni, Rings in which every ideal contained in the set of zero-divisors is a d-ideal, Commun. Korean Math. Soc., 37(1) (2022), 45-56.
N. Ashrafi and M. Pouyan, The unit sum number of Baer rings, Bull. Iranian Math. Soc., 42(2) (2016), 427-434.
A. Barnard, Multiplication modules, J. Algebra, 71(1) (1981), 174-178.
J. Björk, Rings satisfying a minimum condition on principal ideals, J. Reine Angew. Math., 236 (1969), 112-119.
M. Davoudian, Modules with chain condition on non-finitely generated submodules, Mediterr. J. Math., 15(1) (2018), 1 (12 pp).
Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra, 16(4) (1988), 755-779.
M. W. Evans, On commutative P. P. rings, Pacific J. Math., 41 (1972), 687-697.
X. J. Guo and K. P. Shum, Baer semisimple modules and Baer rings, Algebra Discrete Math., 7(2) (2008), 42-49.
E. Houston and M. Zafrullah, Integral domains in which any two $v$-coprime elements are comaximal, J. Algebra, 423 (2015), 93-113.
C. Jayaram, Baer ideals in commutative rings, Indian. J. Pure Appl. Math., 15(8) (1984), 855-864.
C. Jayaram and U. Tekir, von Neumann regular modules, Comm. Algebra, 46(5) (2018), 2205-2217.
C. Jayaram, U. Tekir and S. Koç, Quasi regular modules and trivial extension, Hacet. J. Math. Stat., 50(1) (2021), 120-134.
C. Jayaram, U. Tekir and S. Koç, On Baer modules, Rev. Union Mat. Argentina, 63(1) (2022), 109-128.
J. Jenkins and P. F. Smith, On the prime radical of a module over a commutative ring, Comm. Algebra, 20(12) (1992), 3593-3602.
H. Khabazian, S. Safaeeyan and M. R. Vedadi, Strongly duo modules and rings, Comm. Algebra, 38 (2010), 2832-2842.
T. K. Lee and Y. Zhou, Reduced modules, in: ``Rings, modules, algebras, and abelian groups'', Lecture Notes in Pure and Applied Mathematics, Vol. 236, Dekker, New York, 2004, 365-377.
K. H. Leung and S. H. Man, On commutative Noetherian rings which satisfy the radical formula, Glasgow Math. J., 39(3) (1997), 285-293.
H. Lindo and P. Thompson, The trace property in preenveloping classes, arXiv:2202.03554.
R. L. McCasland and M. E. Moore, On radicals of submodules, Comm. Algebra, 19(5) (1991), 1327-1341.
W. K. Nicholson, J. K. Park and M. F. Yousif, Principally quasi-injective Modules, Comm. Algebra, 27(4) (1999), 1683-1693.
D. Pusat-Yilmaz and P. F. Smith, Modules which satisfy the radical formula, Acta Math. Hungar., 95(1-2) (2002), 155-167.
S. Safaeeyan and A. Taherifar, d-ideals, fd-ideals and prime ideals, Quaest. Math., 42(6) (2019), 717-732.
H. Sharif, Y. Sharifi and S. Namazi, Rings satisfying the radical formula, Acta Math. Hungar., 71(1-2) (1996), 103-108.
T. P. Speed, A note on commutative Baer rings, J. Austral. Math. Soc., 14 (1972), 257-263.
F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications, 22, Springer, Singapore, 2016.