强Pr&uuml;fer环的同调刻画

doi:10.12386/A2021sxxb0028

鏁板瀛︽姤鈥衡�� 2021, Vol. 64 鈥衡�� Issue (2): 311-316.DOI: 10.12386/A2021sxxb0028

鈥� 璁烘枃鈥� 涓婁竴绡� 涓嬩竴绡�

寮篜rüfer鐜殑鍚岃皟鍒荤敾

鐜嬭姵璐�¹, 涔旂¹, 鍛ㄥ痉宸�²

1. 鍥涘窛甯堣寖澶у鏁板绉戝瀛﹂櫌鎴愰兘 610068;
2. 瑗垮崡绉戞妧澶у鐞嗗闄� 缁甸槼 621010

鏀剁鏃ユ湡:2020-03-07 淇洖鏃ユ湡:2020-07-11 鍙戝竷鏃ユ湡:2021-05-15
鍩洪噾璧勫姪:
鍥藉鑷劧绉戝鍩洪噾璧勫姪椤圭洰锛�11671283锛�11701398锛�

A Homological Characterization of Strong Prüfer Rings

Fang Gui WANG¹, Lei QIAO¹, De Chuan ZHOU²

1. School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068, P. R. China;
2. College of Science, Southwest University of Science and Technology, Mianyang 621010, P. R. China

Received:2020-03-07 Revised:2020-07-11 Published:2021-05-15

鎽樿/Abstract

鎽樿锛�

璁�R鏄幆锛�R鐨勫皬finitistic缁存暟瀹氫箟涓篺PD 锛�R锛�=sup{pd_RM|M∈FBR}.鏈枃璇佹槑浜嗭細鑻�R鏄繛閫氱殑寮篜rüfer鐜紝鍒檉PD 锛�R ≤ 1.涔熻瘉鏄庝簡鑻�R鏄己Prüfer鐜紝M∈FBR锛屼笖M鏄�Q-鎸犳ā锛屽垯pd_RM ≤ 1.

鍏抽敭璇�: 鏈夐檺鎶曞皠鍒嗚В, 灏廸initistic缁存暟, Q-鎸犳ā, 寮篜rü, fer鐜�, 杩為�氱幆

Abstract:

Let R be a commutative ring. Then the small finitistic projective dimension of R is defined as fPD(R)=sup{pd_RM|M ∈ FPR}. In this paper, it is shown that if R is a connected strong Prüfer ring, then fPD(R) ≤ 1. It is also shown that if R is a strong Prüfer ring, and if M is a Q-torsion module with M ∈ FPR, then pd_RM ≤ 1.

Key words: finite projective resolution, small finitistic projective dimension, Q-torison module, strong Prü, fer ring, connected ring

涓浘鍒嗙被鍙�:

O154

鐜嬭姵璐�, 涔旂, 鍛ㄥ痉宸�. 寮篜rüfer鐜殑鍚岃皟鍒荤敾[J]. 鏁板瀛︽姤, 2021, 64(2): 311-316.

Fang Gui WANG, Lei QIAO, De Chuan ZHOU. A Homological Characterization of Strong Prüfer Rings[J]. Acta Mathematica Sinica, Chinese Series, 2021, 64(2): 311-316.

0
/ / 鎺ㄨ崘

瀵煎嚭寮曠敤绠＄悊鍣� EndNote|Ris|BibTeX

閾炬帴鏈枃: https://actamath.cjoe.ac.cn/Jwk_sxxb_cn/CN/10.12386/A2021sxxb0028

https://actamath.cjoe.ac.cn/Jwk_sxxb_cn/CN/Y2021/V64/I2/311

鍙傝�冩枃鐚�

[1] Anderson D. D., Anderson D. F., Markanda R., The rings R(x) and R x, J. Algebra, 1985, 95(1):96-115.
[2] Bass H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc., 1960, 95:466-488.
[3] Bazzoni S., Glaz S., Prüfer rings, In:Multiplicative Ideal Theory in Commutative Algebra, Springer, New York, 2006:55-72.
[4] Butts H. S., Smith W., Prüfer rings, Math. Z., 1967, 95:196-211.
[5] Cahen P. J., Fontana M., Frisch S., Glaz S., Open problems in commutative ring theory, In:Commutative Algebra, Springer, New York, 2014:353-375.
[6] Glaz S., Commutative Coherent Rings, Springer-Verlag, Berlin, 1989.
[7] Glaz S., Schwarz R., Prüfer conditions in commutative rings, Arab. J. Sci. Eng., 2011, 36(6):967-983.
[8] Griffin M., Prüfer rings with zero-divisors, J. Reine Angew. Math., 1969, 239/240:55-67.
[9] Huckaba J., Commutative Rings with Zero Divisors, Marcel Dekker Inc., New York, 1988.
[10] Klingler L., Lucas T. G., Sharma T., Local types of Prüfer rings, J. Algebra Appl., 2019, 18(3):1950042, 23 pp.
[11] Li W. X., Chen J. L., Kourki F., On strongly C2 modules and D2 modules, J. Algebra Appl., 2013, 12(7):1350029, 14 pp.
[12] Lucas T. G., Strong Prüfer rings and the ring of finite fractions, J. Pure Appl. Algebra, 1993, 84(1):59-71.
[13] Lucas T. G., The integral closure of R(x) and R x, Comm. Algebra, 1997, 25(3):847-872.
[14] Prüfer H., Untersuchungen über Teilbarkeitseigenschaften in Körpern (in German), J. Reine Angew. Math., 1932, 168:1-36.
[15] Wang F. G., Kim H., Foundations of Commutative Rings and Their Modules, Springer, Singapore, 2016.
[16] Wang F. G., Qiao L., Kim H., Super finitely presented modules and Gorenstein projective modules, Comm. Algebra, 2016, 44(9):4056-4072.
[17] Wang F. G., Kim H., Xiong T., Finitistic weak dimensions of pullbacks, J. Pure Appl. Algebra, 2020, 224(6):106274, 12 pp.
[18] Wang F. G., Zhou D. C., Kim H., Chen D., Module-theoretic characterizations of the ring of finite fractions of a commutative ring, J. Commut. Algebra, 2020, https://projecteuclid.org/euclid.jca/1589335712.

寮篜rüfer鐜殑鍚岃皟鍒荤敾

A Homological Characterization of Strong Prüfer Rings

PDF (PC)

鍙鍖�

鎽樿/Abstract

寮曠敤鏈枃

浣跨敤鏈枃

鍙傝�冩枃鐚�

鐩稿叧鏂囩珷 1

缂栬緫鎺ㄨ崘

Metrics

鏈枃璇勪环

鎵爜鍒嗕韩