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Keywords
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Extending modules, Finite length module, Morphic module, Noetherian module, nonsingular module, Quasi-morphic module
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Abstract
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An R-module M is called quasi-morphic if for any f ∈ End R (M),
there exist g,h ∈ End R (M) such that Imf = Kerg and Kerf = Imh. In addi-
tion, M R 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 coin-
cide. Over a principal ideal domain R, direct sums of cyclic modules which are
(quasi-)morphic are characterized. Among applications of our results, nonsin-
gular 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 o
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