Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SOME FACTORIZATION PROPERTIES OF IDEALIZATION IN COMMUTATIVE RINGS WITH ZERO DIVISORS

  • Received : 2022.05.29
  • Accepted : 2023.12.28
  • Published : 2024.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

We study some factorization properties of the idealization R(+)M of a module M in a commutative ring R which is not necessarily a domain. We show that R(+)M is ACCP if and only if R is ACCP and M satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which R(+)M is a BFR. We also characterize the idealization rings which are UFRs.

Keywords

Acknowledgement

The authors would like to thank the referee for careful reading of the paper and very helpful comments.

References

  1. A. G. Agargun, D. D. Anderson, and S. Valdes-Leon, Factorization in commutative rings with zero divisors. III, Rocky Mountain J. Math. 31 (2001), no. 1, 1-21. https://doi.org/10.1216/rmjm/1008959664
  2. D. D. Anderson and S. Valdes-Leon, Factorization in commutative rings with zero divisors, Rocky Mountain J. Math. 26 (1996), no. 2, 439-480. https://doi.org/10.1216/rmjm/1181072068
  3. D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra 1 (2009), no. 1, 3-56. https://doi.org/10.1216/JCA-2009-1-1-3
  4. M. Axtell, U-factorizations in commutative rings with zero divisors, Comm. Algebra 30 (2002), no. 3, 1241-1255. https://doi.org/10.1081/AGB-120004871
  5. M. Axtell, N. R. Baeth, and J. Stickles, Factorizations in self-idealizations of PIRs and UFRs, Boll. Unione Mat. Ital. 10 (2017), no. 4, 649-670. https://doi.org/10.1007/s40574-016-0107-8
  6. A. Bouvier, Anneaux presimplifiables, Rev. Roumaine Math. Pures Appl. 19 (1974), 713-724.
  7. A. Bouvier, Structure des anneaux a factorisation unique, Publ. Dep. Math. (Lyon) 11 (1974), no. 3, 39-49.
  8. G. W. Chang and D. Smertnig, Factorization in the self-idealization of a PID, Boll. Unione Mat. Ital. (9) 6 (2013), no. 2, 363-377.
  9. D. Frohn, ACCP rises to the polynomial ring if the ring has only finitely many associated primes, Comm. Algebra 32 (2004), no. 3, 1213-1217. https://doi.org/10.1081/AGB120027975
  10. J. A. Huckaba, Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.