Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Codominant dimension of rings and modules


Author: Gary L. Eerkes
Journal: Trans. Amer. Math. Soc. 176 (1973), 125-139
MSC: Primary 16A60
DOI: https://doi.org/10.1090/S0002-9947-1973-0314906-3
MathSciNet review: 0314906
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Expanding Nakayama's original concept of dominant dimension, Tachikawa, Müller and Kato have obtained a number of results pertaining to finite dimensional algebras and more generally, rings and their modules. The purpose of this paper is to introduce and examine a categorically dual notion, namely, codominant dimension. Special attention is given to the question of the relation between the codominant and dominant dimensions of a ring. In particular, we show that the two dimensions are equivalent for artinian rings. This follows from our main result that for a left perfect ring R the dominant dimension of each projective left R-module is greater than or equal to n if and only if the codominant dimension of each injective left R-module is greater than or equal to n. Finally, for computations, we consider generalized uniserial rings and show that the codominant dimension, or equivalently, dominant dimension, is a strict function of the ring's Kupisch sequence.


References [Enhancements On Off] (What's this?)

  • [1] H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466-488. MR 28 #1212. MR 0157984 (28:1212)
  • [2] C. Faith and E. A. Walker, Direct-sum representations of injective modules, J. Algebra 5 (1967), 203-221. MR 34 #7575. MR 0207760 (34:7575)
  • [3] K. R. Fuller, Generalized uniserial rings and their Kupisch series, Math. Z. 106 (1968), 248-260. MR 38 #1118. MR 0232795 (38:1118)
  • [4] -, On indecomposable injectives over artinian rings, Pacific J. Math. 29 (1969), 115-135. MR 40 #186. MR 0246917 (40:186)
  • [5] M. Harada, QF-3 and semi-primary PP-rings. II, Osaka J. Math. 3 (1966), 21-27. MR 34 #5874. MR 0206049 (34:5874)
  • [6] J. P. Jans, Projective injective modules, Pacific J. Math. 9 (1959), 1103-1108. MR. 22 #3750. MR 0112904 (22:3750)
  • [7] T. Kato, Dominant modules, J. Algebra 14 (1970), 341-349. MR 41 #1792. MR 0257138 (41:1792)
  • [8] -, Rings of dominant dimension $ \geqq 1$, Proc. Japan Acad. 44 (1968), 579-584. MR 38 #4525. MR 0236227 (38:4525)
  • [9] -, Rings of U-dominant dimension $ \geqq 1$, Tôhoku Math. J. (2) 21 (1969), 321-327. MR 40 #1423. MR 0248169 (40:1423)
  • [10] H. Kupisch, Beiträge zur Theorie nichtalbeinfacher Ringe mit Minimalbedingung, J. Reine Angew. Math 201 (1959), 100-112. MR 21 #3460. MR 0104707 (21:3460)
  • [11] E. Matlis, Injective modules over noetherian rings, Pacific J. Math. 8 (1958), 511-528. MR 20 #5800. MR 0099360 (20:5800)
  • [12] B. J. Müller, On algebras of dominant dimension one, Nagoya Math. J. 31 (1968), 173-183. MR 36 #2646. MR 0219566 (36:2646)
  • [13] -, The classification of algebras by dominant dimension, Canad. J. Math. 20 (1968), 398-409. MR 37 #255. MR 0224656 (37:255)
  • [14] B. J. Müller, Dominant dimension of semi-primary rings, J. Reine Angew. Math 232 (1968), 173-179. MR 38 #2175. MR 0233854 (38:2175)
  • [15] I. Murase, On the structure of generalized uniserial rings. I, Sci. Papers Coll. Gen. Ed. Univ. Tokyo 13 (1963), 1-22. MR 38 #118. MR 0156875 (28:118)
  • [16] -On the structure of generalized uniserial rings. II, Sci. Papers Coll. Gen. Ed. Univ. Tokyo 13 (1963), 131-158. MR 28 #5086. MR 0161882 (28:5086)
  • [17] -, On the structure of generalized uniserial rings. III, Sci. Papers Coll. Gen. Ed. Univ. Tokyo 14 (1964), 11-25. MR 31 #2277. MR 0178019 (31:2277)
  • [18] T. Nakayama, On algebras with complete homology, Abh. Math. Sem. Univ. Hamburg 22 (1958), 300-307. MR 21 # 3471. MR 0104718 (21:3471)
  • [19] B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373-387. MR 34 # 4305. MR 0204463 (34:4305)
  • [20] H. Tachikawa, A characterization of QF-3 algebras, Proc. Amer. Math. Soc. 13 (1962), 701-703. MR 26 # 5027. MR 0147512 (26:5027)
  • [21] -, On dominant dimensions of QF-3 algebras, Trans. Amer. Math. Soc. 112 (1964), 249-266. MR 28 #5092. MR 0161888 (28:5092)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A60

Retrieve articles in all journals with MSC: 16A60


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0314906-3
Keywords: Perfect rings, minimal injective cogenerator, minimal projective resolutions, injective projective modules, codominant and dominant dimensions, generalized uniserial rings, Kupisch sequences
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society