Research ArticleNo Access

Commutative rings with invertible-radical factorization

Department of Mathematics and Statistics, Hazara University Mansehra, KPK, Pakistan

Corresponding author.

Laboratory of Modeling and Mathematical, Structures Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco

E-mail Address: mahdou@hotmail.com

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, and

Youssef Zahir

Department of Mathematics, Faculty of Sciences of Rabat, B. P. 1014, Mohammed V University in Rabat, Morocco

E-mail Address: y.zahir@um5r.ac.ma

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https://doi.org/10.1142/S0219498822501535Cited by:1 (Source: Crossref)

Abstract

In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties under homomorphic image and their transfer to various contexts of constructions such as direct product, trivial ring extension and amalgamated duplication of a ring along an ideal. Our results generate examples that enrich the current literature with new and original families of rings satisfying these properties.

Communicated by E. Kwak

Keywords:

AMSC: 13B99, 13A15, 13G05, 13B21

References

1. M. T. Ahmed , SP-domains are almost Dedekind a streamlined proof, Discuss. Math. Gen. Algebra Appl. 40(1) (2020) 21–24. Crossref, Google Scholar
2. M. T. Ahmed and T. Dumitrescu , Domains with invertible-radical factorization, Bull. Korean Math. Soc. 55(3) (2018) 809–818. Web of Science, Google Scholar
3. M. T. Ahmed and T. Dumitrescu , SP-rings with zero-divisors, Comm. Algebra 45(10) (2017) 4435–4443. Crossref, Web of Science, Google Scholar
4. M. M. Ali , Multiplication Modules and Homogeneous Idealization II, Beitr, Algebra Geom. 48(2) (2007) 321–343. Google Scholar
5. D. D. Anderson and M. Winders , Idealization of a module, J. Commut. Algebra 1(1) (2009) 3–56. Crossref, Web of Science, Google Scholar
6. C. Bakkari, S. Kabbaj and N. Mahdou , Trivial extensions defined by Prüfer conditions, J. Pure App. Algebra 214(1) (2010) 53–60. Crossref, Web of Science, Google Scholar
7. M. Chhiti and N. Mahdou , Some homological properties of amalgamated duplication of a ring along an ideal, Bull. Iranian Math. Soc. 38(2) (2012) 507–515. Web of Science, Google Scholar
8. M. D’Anna , A construction of Gorenstein rings, J. Algebra 306 (2006) 507–519. Crossref, Web of Science, Google Scholar
9. M. D’Anna and M. Fontana , An amalgamated duplication of ring along an ideal: The basic properties, J. Algebra Appl. 6(3) (2007) 443–459. Link, Web of Science, Google Scholar
10. R. Gilmer , Multiplicative Ideal Theory (Marcel Dekker, New York, 1972). Google Scholar
11. S. Glaz , Commutative Coherent Rings, Lecture Notes in Mathematics, Vol. 1371 (Springer-Verlag, 1989). Crossref, Google Scholar
12. M. Griffin , Prüfer rings with zero divisors, J. Reine Angew. Math. 239/240 (1969) 55–67. Web of Science, Google Scholar
13. J. A. Huckaba , Commutative Rings with Zero Divisors (Marcel Dekker, New York Basel, 1988). Google Scholar
14. S. Kabbaj and N. Mahdou , Trivial extensions defined by coherent-like conditions, Comm. Algebra 32(10) (2004) 3937–3953. Crossref, Web of Science, Google Scholar
15. M. Larsen , A generalization of almost Dedekind domains, J. Reine Angew. Math. 245 (1970) 119–123. Google Scholar
16. T. G. Lucas , Additively regular rings and Marot rings, Palest. J. Math. 5 (2016) 90–99. Web of Science, Google Scholar
17. J. Mott , Multiplication rings containing only finitely many minimal prime ideals, J. Sci. Hiroshima Univ. Ser. A-I 33 (1969) 73–83. Google Scholar
18. M. Nagata , Local Rings (Wiley-Interscience, New York, 1962). Google Scholar
19. B. Olberding , Factorization into radical ideals, in Arithmetical Properties of Commutative Rings and Monoids, ed. S. Chapman , Lecture Notes in Pure Applied Mathematics, Vol. 241 (Chapman Hall, Boca Raton), pp. 363–377. Google Scholar
20. B. Olberding , Factorization into prime and invertible ideals, J. London Math. Soc. 62 (2000) 336–344. Crossref, Web of Science, Google Scholar
21. B. Olberding , Factorization into prime and invertible ideals II, J. London Math. Soc. 80 (2009) 155–170. Crossref, Web of Science, Google Scholar
22. N. Vaughan and R. Yeagy , Factoring ideals into semiprime ideals, Can. J. Math. 30 (1978) 1313–1318. Crossref, Web of Science, Google Scholar