| On Maximal Injectivity |
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| 作者姓名： | Ming Yi WANG Guo ZHAO |
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| 作者单位： | [1]Institute of Mathematics, Sichuan Normal University, Chengdu 610066, P. R. China [2]College of Computer Science and Technology, Southwest University for Nationalities, Chengdu 610041, P. R. China |
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| 基金项目： | Supported by the Natural Science Foundation of Education Department of Sichuan Province, and the Youth Science Foundation of Sichuan Province |
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| 摘 要： | A right R-module E over a ring R is said to be maximally injective in case for any maximal right ideal m of R, every R-homomorphism f ： m → E can be extended to an R-homomorphism f^1 ： R → E. In this paper, we first construct an example to show that maximal injectivity is a proper generalization of injectivity. Then we prove that any right R-module over a left perfect ring R is maximally injective if and only if it is injective. We also give a partial affirmative answer to Faith＇s conjecture by further investigating the property of maximally injective rings. Finally, we get an approximation to Faith＇s conjecture, which asserts that every injective right R-module over any left perfect right self-injective ring R is the injective hull of a projective submodule.
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| 关 键 词： | 最大内射性 准Frobenius环 Faith猜想 内射环 |
| 收稿时间： | 2004-03-29 |
| 修稿时间： | 2004-03-292004-08-16 |
On Maximal Injectivity |
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| Ming Yi WANG Guo ZHAO.On Maximal Injectivity[J].Acta Mathematica Sinica,2005,21(6):1451-1458. |
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| Authors: | Ming Yi Wang Guo Zhao |
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| Institution: | (1) Institute of Mathematics, Sichuan Normal University, Chengdu 610066, P. R. China;(2) College of Computer Science and Technology, Southwest University for Nationalities, Chengdu 610041, P. R. China |
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| Abstract: | A right R–module E over a ring R is said to be maximally injective in case for any maximal right ideal m of R, every R–homomorphism f : m → E can be extended to an R–homomorphism f' : R → E. In this paper, we first construct an example to show that maximal injectivity is a proper generalization of injectivity.
Then we prove that any right R–module over a left perfect ring R is maximally injective if and only if it is injective. We also give a partial affirmative answer to Faith's conjecture by
further investigating the property of maximally injective rings. Finally, we get an approximation to Faith's conjecture, which
asserts that every injective right R–module over any left perfect right self–injective ring R is the injective hull of a projective submodule.
Supported by the Natural Science Foundation of Education Department of Sichuan Province, and the Youth Science Foundation
of Sichuan Province |
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| Keywords: | Maximal injectivity Quasi– Frobenius rings Faith's conjecture |
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