On strongly Gorenstein hereditary rings

다국어 초록 (Multilingual Abstract)

In this note, we mainly discuss strongly Gorenstein hereditary rings. We prove that for any ring, the class of $SG$-projective modules and the class of $G$-projective modules coincide if and only if the class of $SG$-projective modules is closed under extension. From this we get that a ring is an $SG$-hereditary ring if and only if every ideal is $G$-projective and the class of $SG$-projective modules is closed under extension. We also give some examples of domains whose ideals are $SG$-projective.

참고문헌 (Reference)

1 G. Zhao, "n-strongly Gorenstein projective, injective and at modules" 39 (39): 3044-3062, 2011

2 L. Salce, "Wareld domains: module theory from linear algebra to commutative algebra through abelian groups" 70 : 163-185, 2009

3 E. Matlis, "The two-generator problem for ideals" 17 : 257-265, 1970

4 N. Mahdou, "Strongly n-Gorenstein projective, injective, and at modules" 87 (87): 35-53, 2018

5 D. Bennis, "Strongly Gorenstein projective, injective, and at modules" 210 (210): 437-445, 2007

6 X. Yang, "Strongly Gorenstein projective, injective and at modules" 320 (320): 2659-2674, 2008

7 K. Hu, "Some results on Gorenstein Dedekind domains and their factor rings" 41 (41): 284-293, 2013

8 Kui Hu, "On overrings of Gorenstein Dedekind domains" 대한수학회 50 (50): 991-1008, 2013

9 N. Mahdou, "On (strongly) Gorenstein (semi) hereditary rings" 36 (36): 431-440, 2011

10 E. E. Enochs, "Gorenstein injective and projective modules" 220 (220): 611-633, 1995

11 H. Holm, "Gorenstein homological dimensions" 189 (189): 167-193, 2004

12 D. Bennis, "Gorenstein analogue of Auslander's theorem on the global dimension" 43 (43): 174-181, 2015

13 F. Wang, "Foundations of Commutative Rings and their Modules" Springer 2016

14 I. Kaplansky, "Commutative Rings" The University of Chicago Press 1974

15 H. Bass, "Big projective modules are free" 7 : 24-31, 1963

16 J. J. Rotman, "An Introduction to Homological Algebra" Springer 2009