THE FIRST U-EXTENSION MODULE AS CLASSES OF SHORT U-EXACT SEQUENCES

Yudi Mahatma

Abstract


Inspired by the notions of the U-exact sequence introduced by Davvaz and Parnian-Garamaleky in 1999, and of the chain U-complex introduced by Davvaz and Shabani-Solt in 2002, Mahatma and Muchtadi-Alamsyah in 2017 developed the concept of the U-projective resolution and the U-extension module, which are the generalizations of the concept of the projective resolution and the concept of extension module, respectively. It is already known that every element of a first extension module can be identified as a short exact sequence. To the simple, there is a relation between the first extension module and the short exact sequence. It is proper to expect the relation to be provided in the U-version. In this paper, we aim to construct a one-one correspondence between the first U-extension module and the set consisting of equivalence classes of short U-exact sequence.

Keywords: Chain U-complex, U-projective resolution, U-extension module


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References


Davvaz, B., Shabani-Solt, H., 2002, A generalization of homological algebra, J. Korean Math. Soc. No. 6, 39 : 891 – 898

Davvaz, B., Parnian-Garamaleky, A., 1999, A note on exact sequences, Bull. Malaysian Math. Soc. (Second Series), 22 : 53 – 56

Mahatma, Y., Muchtadi-Alamsyah, I., 2017, Construction of U-extension module, AIP Conference Proceedings, 1867 : 020025-1 – 020025-9

Rotman, J. J., 2009, Introduction to Homological Algebra, Edisi ke-2, Springer, New York.

Baur, K., Mahatma, Y., Muchtadi-Alamsyah, I., 2019, The U-projective resolution of modules over path algebras of type An and An, Communications of the Korean Mathematical Society, 34 : 701 – 718




DOI: https://doi.org/10.25077/jmu.10.4.553-560.2021

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