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Equivalence of diagonal matrices over local rings. (English) Zbl 1158.16011

The article uses two equivalence relations of finitely presented modules:
(1) epigeny, when two modules are equivalent if and only if either of them is an epimorphic image of the other;
(2) lower part, which is a dual of epigeny in a certain sense.
The main result (Theorem 5.3) is a Krull-Schmidt type theorem: two finite direct sums of non-zero cyclic modules over local rings are isomorphic if and only if both the epigeny classes and the lower parts of the summands of the sums are the same, counted with multiplicity. However, the pairing of epigeny classes and lower parts induced by the summands may differ in the two sums.
The proof uses typical arguments over local rings, based on similar results on uniserial modules. A notable aspect is that the endomorphism ring of cyclic modules is semi-local with at most two maximal ideals (Theorem 2.1), which correspond to the epigeny class and lower part.
The authors claim without proof that their results extend to locally presented modules and couniformly presented modules over semi-perfect rings.

MSC:

16L30 Noncommutative local and semilocal rings, perfect rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16S50 Endomorphism rings; matrix rings
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