The large condition for rings with Krull dimension

Author:
Ann K. Boyle

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

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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 is strictly less than the Krull dimension of *M*. For a right noetherian ring *R* with Krull dimension this is equivalent to the condition that every f.g. uniform submodule of with Krull dimension is critical. It is also shown that if *R* is right noetherian with Krull dimension and if is a right ideal maximal with respect to *K* , then *R* satisfies the large condition if and only if is a finite intersection of cocritical right ideals and is closed.

**[1]**A. K. Boyle and E. H. Feller,*Semicritical modules and k-primitive rings*, Module Theory, Selected Papers and Problems, Seattle 1977, Lecture Notes in Math., Springer-Verlag, Berlin and New York (to appear). MR**550429 (81c:16038)****[2]**A. K. Boyle, M. G. Deshpande and E. H. Feller,*On nonsingularly k-primitive rings*, Pacific J. Math.**68**(1977), 303-311. MR**0457491 (56:15696)****[3]**M. G. Deshpande,*Structure of right subdirectly irreducible rings*. I, J. Algebra**17**(1971), 317-325. MR**0274491 (43:254)****[4]**A. W. Goldie,*Semi-prime rings with maximum condition*, Proc. London Math. Soc.**10**(1960), 201-220. MR**0111766 (22:2627)****[5]**K. R. Goodearl,*Ring theory*, Dekker, New York, 1976. MR**0429962 (55:2970)****[6]**R. Gordon and J. C. Robson,*Krull dimension*, Mem. Amer. Math. Soc. No. 133 (1973). MR**0352177 (50:4664)****[7]**D. W. Sharpe and P. Vamos,*Injective modules*, Cambridge Univ. Press, Cambridge, 1972. MR**0360706 (50:13153)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1978-0503524-X

Keywords:
The large condition,
semicritical module,
semiprime ring,
nonsingularly *k*-primitive ring

Article copyright:
© Copyright 1978
American Mathematical Society