NOTE ON THE DW RINGS

Abstract

In this paper we are mainly concerned with DW rings, i.e., rings

in which every ideal is a w-ideal. We give some new classes of DW rings and we

show how the concept of DW domains is used to characterize Prufer domains

and Dedekind domains. Namely, we prove that a ring is a Prufer domain

(resp., Dedekind domain) if and only if it a coherent (resp., Noetherian) DW

domain with nite weak global dimension.

Keywords

• DW ring and domain,PvMD,Krull domain

References

1. N. Q. Ding and J. L. Chen, The flat dimensions of injective modules, Manuscripta Math., 78(2) (1993), 165-177.
2. K. Fujita, Innite dimensional Noetherian Hilbert domains, Hiroshima Math. J., 5(2) (1975), 181-185.
3. S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics, 1371, Springer-Verlag, Berlin, 1989.
4. S. Glaz and W. V. Vasconcelos, Flat ideals. II, Manuscripta Math., 22(4) (1977), 325-341.
5. J. R. Hedstrom and E. G. Houston, Some remarks on star-operations, J. Pure Appl. Algebra, 18(1) (1980), 37-44.
6. W. Heinzer, Integral domains in which each non-zero ideal is divisorial, Mathematika, 15 (1968), 164-170.
7. H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189(1-3) (2004), 167-193.
8. E. G. Houston and M. Zafrullah, Integral domains in which each t-ideal is divisorial, Michigan Math. J., 35(2) (1988), 291-300.

9. E. G. Houston, S. Kabbaj and A. Mimouni, ?-Reductions of ideals and Prufer v-multiplication domains, J. Commut. Algebra, 9(4) (2017), 491-505.
10. B. G. Kang, Prufer v-multiplication domains and the ring R[X]Nv , J. Algebra, 123(1) (1989), 151-170.
11. I. Kaplansky, Commutative Rings, Revised Edition, Univ. Chicago Press, Chicago, 1974.
12. H. Kim and F. G. Wang, On LCM-stable modules, J. Algebra Appl., 13(4) (2014), 1350133 (18 pp).
13. N. Mahdou and M. Tamekkante, On (weak) Gorenstein global dimensions, Acta Math. Univ. Comenian. (N.S.), 82(2) (2013), 285-296.
14. N. Mahdou, M. Tamekkante and S. Yassemi, On (strongly) Gorenstein von Neumann regular rings, Comm. Algebra, 39(9) (2011), 3242-3252.
15. A. Mimouni, TW-domains and strong Mori domains, J. Pure Appl. Algebra, 177(1) (2003), 79-93.
16. A. Mimouni, Integral domains in which each ideal is a w-ideal, Comm. Algebra, 33(5) (2005), 1345-1355.
17. F. G. Wang, On w-projective modules and w- at modules, Algebra Colloq., 4(1) (1997), 111-120.
18. F. G. Wang, Finitely presented type modules and w-coherent rings, J. Sichuan Normal Univ., 33 (2010), 1-9.
19. F. G. Wang and H. Kim, w-Injective modules and w-semi-hereditary rings, J. Korean Math. Soc., 51(3) (2014), 509-525.
20. F. G. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications, 22, Springer, Singapore, 2016.
21. F. G. Wang and R. L. McCasland, On w-modules over strong Mori domains, Comm. Algebra, 25(4) (1997), 1285-1306.
22. F. G. Wang and R. L. McCasland, On strong Mori domains, J. Pure Appl. Algebra, 135(2) (1999), 155-165.
23. F. G. Wang and L. Qiao, The w-weak global dimension of commutative rings, Bull. Korean Math. Soc., 52(4) (2015), 1327-1338.
24. F. G. Wang and J. Zhang, Injective modules over w-Noetherian rings, Acta Math. Sinica (Chin. Ser.), 53(6) (2010), 1119-1130 (in Chinese).
25. H. Yin, F. G. Wang, X. Zhu and Y. Chen, w-Modules over commutative rings, J. Korean Math. Soc., 48(1) (2011), 207-222.
26. M. Zafrullah, The v-operation and intersection of quotient of integral domains, Comm. Algebra, 13(8) (1985), 1699-1712.
27. S. Q. Zhao, F. G. Wang and H. L. Chen, Flat modules over a commutative ring are w-modules, J. Sichuan Normal Univ., 35 (2012), 364-366 (in Chinese).