Abstract
Let R be a commutative ring with unity, M be a unitary R-module and G a finite abelian group (viewed as a \({\mathbb {Z}}\)-module). The main objective of this paper is to study properties of mod-annihilators of M. For \(x \in M\), we study the ideals \([x : M] =\{r\in R ~|~ rM\subseteq Rx\}\) of R corresponding to mod-annihilator of M. We investigate as when [x : M] is an essential ideal of R. We prove that the arbitrary intersection of essential ideals represented by mod-annihilators is an essential ideal. We observe that [x : M] is injective if and only if R is non-singular and the radical of R/[x : M] is zero. Moreover, if essential socle of M is non-zero, then we show that [x : M] is the intersection of maximal ideals and \([x : M]^2 = [x : M]\). Finally, we discuss the correspondence of essential ideals of R and vertices of the annihilating graphs realized by M over R.
Similar content being viewed by others
References
Aijaz, M., Pirzada, S., Somasundaram, A.: Zero divisor graphs of unitary R-modules over commutative rings. AKCE Int. J. Graphs Combin. (in press)
Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)
Anderson, D.F., Levy, R., Shapiro, J.: Zero-divisor graphs, von Neumann regular rings, and Boolean algebras. J. Pure Appl. Algebra 180, 221–241 (2003)
Azarpanah, F.: Essential ideals in C(X). Period. Math. Hung. 31, 105–112 (1995)
Azarpanah, F.: Intersection of essential ideals in C(X). Proc. Am. Math. Soc. 125, 2149–2154 (1997)
Beck, I.: Coloring of commutative rings. J. Algebra 116, 208–226 (1988)
Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press, Princeton (1956)
Green, B.W., Van Wyk, L.: On the small and essential ideals in certain classes of rings. J. Aust. Math. Soc. Ser. A 46, 262–271 (1989)
Hongin, F., Stewart, P.: Graded rings and essential ideals. Acta Math. Sin. 9(4), 344–351 (1993)
Jain, S.K., Lopez-permouth, S.R., Rizvi, S.T.: Continuous rings with ACC on essentials is left Artinian. Proc. Am. Math. Soc. 108(3), 192–195 (1990)
Kaneda, M., Paulsen, V.I.: Characterization of essential ideals as operator modules over \(C^{*}-algebras\). J. Oper. Theory 49, 245–262 (2003)
Karamzadeh, O.A.S., Rostami, M.: On the intrinsic topology and some related ideals of C(X). Proc. Am. Math. Soc. 93(1), 179–184 (1985)
Meyer, F.D., Meyer, L.D.: Zero-divisor graphs of semigroups. J. Algebra 283, 190–198 (2005)
Momathan, E.: Essential ideals in rings of measurable functions. Commun. Algebra 38, 4739–4746 (2010)
Pirzada, S., Rather, A Bilal, Chishti, T.A.: On distance Laplacian spectrum of zero divisor graphs of \({\mathbb{Z}}_{n}\). Carpathian Math. Publ. 13(1), 48–57 (2021)
Pirzada, S., Wani, Bilal A., Somasundaram, A.: On the eigenvalues of zero divisor graph associated to finite commutative ring \(Z_{p^{M}q^{N}}\). AKCE Int. J. Graphs Combin. 18(1), 1–16 (2021)
Pirzada, S., Raja, Rameez: On graphs associated with modules over commutative rings. J. Korean Math. Soc. 53(5), 1167–1182 (2016)
Pirzada, S., Raja, Rameez: On the metric dimension of a zero-divisor graph. Commun. Algebra 45(4), 1399–1408 (2017)
Raja, Rameez, Pirzada, S.: On annihilating graphs associated with modules over commutative rings. Algebra Colloq. 29(2) (2022) (in press)
Pirzada, S., Aijaz, M., Imran Bhat, M.: On zero divisor graphs of the rings \(Z_{n}\). Afr. Mat. 31, 727–737 (2020)
Puczylowski, E.R.: On essential extensions of rings. Bull. Aust. Math. Soc. 35, 379–386 (1987)
Rather, B.A., Pirzada, S., Naikoo, T.A., Shang, Y.: On Laplacian eigenvalues of the zero-divisor graph associated to the ring of integers modulo n. Mathematics 9(482), 1–17 (2021)
Redmond, S.P.: The zero-divisor graph of a non-commutative ring. Int. J. Commut. Rings 1(4), 203–211 (2002)
Resenberg, A., Zelinsky, D.: Finiteness of the injective hull. Math. Z. 70, 372–380 (1959)
Wisbauer, R.: Foundations of Modules and Ring Theory. Gordon and Breach, Reading (1991)
Acknowledgements
The authors are grateful to the anonymous referees for their valuable suggestions which improved the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Raja, R., Pirzada, S. Essential ideals represented by mod-annihilators of modules. Afr. Mat. 33, 52 (2022). https://doi.org/10.1007/s13370-022-00988-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13370-022-00988-9