The large condition for rings with Krull dimension
HTML articles powered by AMS MathViewer
- by Ann K. Boyle PDF
- Proc. Amer. Math. Soc. 72 (1978), 27-32 Request permission
Abstract:
A module M with Krull dimension is said to satisfy the large condition if for any essential submodule L of M, the Krull dimension of $M/L$ is strictly less than the Krull dimension of M. For a right noetherian ring R with Krull dimension $\alpha$ this is equivalent to the condition that every f.g. uniform submodule of $E({R_R})$ with Krull dimension $\alpha$ is critical. It is also shown that if R is right noetherian with Krull dimension $\alpha$ and if ${I_0}$ is a right ideal maximal with respect to K $\dim {I_0} < \alpha$, then R satisfies the large condition if and only if ${I_0}$ is a finite intersection of cocritical right ideals and ${I_0}$ is closed.References
- A. K. Boyle and E. H. Feller, Semicritical modules and $k$-primitive rings, Module theory (Proc. Special Session, Amer. Math. Soc., Univ. Washington, Seattle, Wash., 1977) Lecture Notes in Math., vol. 700, Springer, Berlin, 1979, pp. 57–74. MR 550429
- A. K. Boyle, M. G. Deshpande, and E. H. Feller, On nonsingularly $k$-primitive rings, Pacific J. Math. 68 (1977), no. 2, 303–311. MR 457491
- Madhukar G. Deshpande, Structure of right subdirectly irreducible rings. I, J. Algebra 17 (1971), 317–325. MR 274491, DOI 10.1016/0021-8693(71)90014-7
- A. W. Goldie, Semi-prime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201–220. MR 111766, DOI 10.1112/plms/s3-10.1.201
- K. R. Goodearl, Ring theory, Pure and Applied Mathematics, No. 33, Marcel Dekker, Inc., New York-Basel, 1976. Nonsingular rings and modules. MR 0429962
- Robert Gordon and J. C. Robson, Krull dimension, Memoirs of the American Mathematical Society, No. 133, American Mathematical Society, Providence, R.I., 1973. MR 0352177
- D. W. Sharpe and P. Vámos, Injective modules, Cambridge Tracts in Mathematics and Mathematical Physics, No. 62, Cambridge University Press, London-New York, 1972. MR 0360706
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 72 (1978), 27-32
- MSC: Primary 16A55
- DOI: https://doi.org/10.1090/S0002-9939-1978-0503524-X
- MathSciNet review: 503524