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Diagonal matrix reduction over Hermite rings

ABSTRACT A commutative ring R is an elementary divisor ring if every matrix over R admits a diagonal reduction. In this paper, we define the term ‘Zabvasky subset’ of a ring to study diagonal matrix reduction. Let S be a Zabavsky subset of a Hermite ring R. We prove that R is an elementary divisor ring if and only if with implies that there exist such that . If with implies that there exists a such that , then R is an elementary divisor ring. Many known results are thereby generalized to much wider class of rings.