Abstract
This paper introduce and study 1-absorbing prime ideals and weakly 1-absorbing prime ideals in noncommutative rings a new class of ideals between prime ideals (weakly prime ideals) and 2-absorbing ideals (weakly 2-absorbing ideals). Let R be a noncommutative ring with a nonzero identity \(1\ne 0\). A proper ideal P of R is said to be a 1-absorbing prime ideal (weakly 1-absorbing prime ideal) if for all nonunits \(x,y,z\in R\) with xRyRz \(\subset \) P \((\left\{ 0\right\} \ne \) xRyRz \(\subset \) P) , then \(xy\in P\) or \(z\in P\). In addition to give many properties and characterizations of these ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. We show that weakly 1-absorbing prime ideals are really of interest in decomposable rings.
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References
Anderson, D.F., Smith, E.: Weakly prime ideals. Houston J. Math. 29(4), 831–840 (2003)
Badawi, A.: On \(2-\)absorbing ideals in commutative rings. Bull. Aust. Math. Soc. 75, 417–429 (2007)
Badawi, A., Darani, A.Y.: On weakly 2-absorbing ideals of commutative rings. Houston J. Math 39(2), 441–452 (2013)
Yassine, A., Nikmehr, M.J., Nikandish, R.: On \(1\)-absorbing prime ideals of commutative rings. J. Algebra Appl. 20, 2150175 (2020)
Koç, S., Tekir, Ü., Yıldız, E.: On weakly \(1\)-absorbing ideals, Ricerche di Matematica, (2021)
Abouhalaka, A., Findik, S.: Almost prime ideals in noncommutative rings, arXiv:2201.09758
Hirano, Y., Poon, E., Tsutsui, H.: On rings in which every ideal is weakly prime. Bull. Korean Math. Soc. 47(5), 1077–1087 (2010)
Groenewald, N.J.: On \(2-\)absorbing ideals in noncommutative rings. JP J. Algebr. Number Theory Appl. 40(5), 855–867 (2018)
Nico, Groenewald: On weakly 2-absorbing ideals of noncommutative rings. Afrika Matematika 32, 1669–1683 (2021)
Nikmehr M.J., Nikandish R., Yassine A.: On weakly \(1\)-absorbing prime ideals of commutative rings, preprint arXiv:2102.06077. 2021
Badawi, A., Tekir, U., Yetkin, E.: On weakly 2-absorbing primary ideals of commutative rings. J. Korean Math. Soc. 52(1), 97–111 (2015)
Birkenmeier, G., Heatherly, H., Lee, E.: Prime ideals and prime radicals in near-rings. Monatshefte für Mathematik 117, 179–197 (1994)
Groenewald, N.J.: On weakly prime and weakly completely prime ideals in noncommutative rings. Int. Electron. J. Algebra 28, 43–60 (2020)
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Communicated by Sergio R. López-Permouth.
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Groenewald, N. 1-absorbing prime ideals and weakly 1-absorbing prime ideals in noncommutative rings. São Paulo J. Math. Sci. 17, 871–887 (2023). https://doi.org/10.1007/s40863-022-00348-2
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DOI: https://doi.org/10.1007/s40863-022-00348-2