Pseudo-irreducible elements in multiplicative lattices

Document Type : Original Paper


1 Department of Pure Mathematics Shahid Bahonar University of Kerman , Kerman, Iran

2 Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran


In this paper, we define the concept of pseudo-irreducible elements in multiplicative lattices and examine its relationship with other important concepts of multiplicative lattices such as prime elements, primary elements, and maximum elements, and then with the help of this concept, we define and analyze the complete comaximal factorization in multiplicative lattices. In particular, we characterize multiplicative lattices whose every element can be written as a product of pseudo-irreducible and pairwise comaximal elements.


Main Subjects

[1] Anderson, D.D. and Jayaram, C., 1995. Regular lattices. Studia Scientiarum Mathematicarum Hungarica, 30(3), pp.379–388.
[2] Alarcon, F., Anderson, D.D. and Jayaram, C., 1995. Some results on abstract commutative ideal theory.Periodica Mathematica Hungarica, 30(1), pp.1–26. doi: 10.1007/bf01876923
[3] Brewer, J.W. and Heinzer, W.J., 2002. On decomposing ideals into products of comaximal ideals.Communications in Algebra, 30(12), pp.5999–6010. doi: 10.1081/agb-120016028
[4] Dumitrescu, T. and Epure, M., 2022. Comaximal factorization lattices. Communications in Algebra, 50(9), pp.4024–4031. doi: 10.1080/00927872.2022.2057512
[5] Gillman, L. and Jerison, M., 1960. Rings of continuous functions. Van Nostrand, Princeton.
[6] Iorgulescu, A., 2004. Classes of BCK algebras-Part III. Preprint series of the Institute of Mathematics of the Romanian Academy, 3, pp.1–37.
[7] Juett, J., 2012. Generalized comaximal factorization of ideals. Journal of Algebra, 352(1), pp.141 166.doi:10.1016/j.jalgebra.2011.11.008
[8] Galatos, N., Jipsen, P., Kowalski, T. and Ono, H., 2007. Residuated lattices: an algebraic glimpse at substructural logics. Elsevier.
[9] Noether, E., 1921. Idealtheorie in ringbereichen. Mathematische Annalen, 83(1-2), pp.24–66. doi:10.1007/bf01464225
[10] Thakare, N.K., Manjarekar, C.S. and Maeda, S., 1988. Abstract spectral theory II: minimal characters and minimal spectrums of multiplicative lattices. Acta Sci. Math, 52, pp.53–67.
[11] Ward, M. and Dilworth, R.P., 1939. Residuated lattices. Trans. Am. Math. Soc., 45, pp.335–354. doi:10.1090/S0002-9947-1939-1501995-3