On quasi-morphic modules

An R-module M is called quasi-morphic if for any f∈EndR(M), there exist g,h∈EndR(M) such that Imf=Kerg and Kerf=Imh. In addition, MR is said to be morphic whenever g=h in the above definition. The main objective of this paper is investigating quasi-morphic property for several classes of modules. Fi...

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Bibliographic Details
Published in:Journal of algebra Vol. 636; pp. 757 - 778
Main Author: Dehghani, Najmeh
Format: Journal Article
Language:English
Published: Elsevier Inc 15.12.2023
Subjects:
Extending module
Finite length module
Morphic module
Noetherian module
Nonsingular module
Quasi-morphic module
secondary
Noetherian module
Quasi-morphic module
Morphic module
Nonsingular module
Finite length module
Extending module
primary
ISSN:0021-8693, 1090-266X
Online Access:Get full text
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Summary:An R-module M is called quasi-morphic if for any f∈EndR(M), there exist g,h∈EndR(M) such that Imf=Kerg and Kerf=Imh. In addition, MR is said to be morphic whenever g=h in the above definition. The main objective of this paper is investigating quasi-morphic property for several classes of modules. First we obtain general properties of quasi-morphic modules via exact sequence approach. Moreover, we investigate conditions under which a finite length quasi-morphic module is morphic. As a result, we show that for uniserial finite length modules, the notions of morphic and quasi-morphic coincide. Over a principal ideal domain R, direct sums of cyclic modules which are (quasi-)morphic are characterized. Among applications of our results, nonsingular extending (quasi-)morphic modules are characterized completely. We also prove that over a commutative Noetherian domain R which is not a field, quasi-morphic nonsingular extending modules are precisely direct sums of copies of Q (the quotient field of R). (Quasi-)Morphic singular extending abelian groups are also characterized.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2023.09.010