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Characterizations of Noetherian and hereditary rings


Author: Zheng-Xu He
Journal: Proc. Amer. Math. Soc. 93 (1985), 414-416
MSC: Primary 16A52; Secondary 16A33, 18G05
DOI: https://doi.org/10.1090/S0002-9939-1985-0773992-0
MathSciNet review: 773992
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Abstract: We characterize the left Noetherian rings by the existence of decompositions of left modules into direct sums of an injective submodule and a submodule containing no injective submodule except 0. We also prove that a left Noetherian ring is left hereditary iff the suspension of each left ideal (see [7]) is injective or, equivalently, the above decomposition is unique.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0773992-0
Keywords: Injective module, $ i$-reduced module, $ i$-decomposition, unique $ i$-decomposition, Noetherian ring, left hereditary ring, FP-injective module, suspension of a module
Article copyright: © Copyright 1985 American Mathematical Society