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.
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Supported by the Natural Science Foundation of Education Department of Sichuan Province, and the Youth Science Foundation of Sichuan Province
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Wang, M.Y., Zhao, G. On Maximal Injectivity. Acta Math Sinica 21, 1451–1458 (2005). https://doi.org/10.1007/s10114-005-0599-0
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DOI: https://doi.org/10.1007/s10114-005-0599-0