Rocky Mountain Journal of Mathematics

On weakly coherent rings

Abstract

In this paper, we define weakly coherent rings and examine the transfer of this property to homomorphic images, trivial ring extensions, localizations and finite direct products. These results provide examples of weakly coherent rings that are not coherent rings. We show that the class of weakly coherent rings is not stable under localization. Also, we show that the class of weakly coherent rings and the class of strongly $2$-coherent rings are not comparable.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 3 (2014), 743-752.

Dates
First available in Project Euclid: 28 September 2014

https://projecteuclid.org/euclid.rmjm/1411945662

Digital Object Identifier
doi:10.1216/RMJ-2014-44-3-743

Mathematical Reviews number (MathSciNet)
MR3264480

Zentralblatt MATH identifier
1301.13023

Citation

Bakkari, Chahrazade; Mahdou, Najib. On weakly coherent rings. Rocky Mountain J. Math. 44 (2014), no. 3, 743--752. doi:10.1216/RMJ-2014-44-3-743. https://projecteuclid.org/euclid.rmjm/1411945662

References

• N. Bourbaki, Commutative algebra, Chapters 1-7, Springer, Berlin, 1998.
• D. Costa, Parameterizing families of non-Noetherian rings, Comm. Alg. \bf22 (1994), 3997-4011.
• R. Gilmer, Multiplicative ideal theory, Pure Appl. Math. \bf12, Marcel Dekker, Inc., New York, 1972.
• S. Glaz, Finite conductor rings, Proc. Amer. Math. Soc. \bf129 (2000), 2833-2843.
• S. Glaz, Commutative coherent rings, Lect. Notes Math. \bf1371, Springer-Verlag, Berlin, 1989.
• –––, Controlling the zero-Divisors of a commutative ring, Lect. Notes Pure Appl. Math. \bf231, Dekker, New York, 2003.
• J.A. Huckaba, Commutative rings with zero divisors, Marcel Dekker, New York, 1988.
• S. Kabbaj and N. Mahdou, Trivial extensions of local rings and a conjecture of Costa, Lect. Notes Pure Appl. Math. \bf231, Dekker, New York, 2003.
• S. Kabbaj and N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Alg. \bf32 (2004), 3937-3953.
• N. Mahdou, On Costa's conjecture, Comm. Alg. \bf29 (2001), 2775-2785.
• –––, On $2$-Von Neumann regular rings, Comm. Alg. \bf33 (2005), 3489-3496.
• J.J. Rotman, An introduction to homological algebra, Academic Press, New York, 1979. \noindentstyle