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Fine elements were defined in  for unital rings, as sums of units by nilpotents. Such a sum is called uniquely fine if it is the only decomposition t u + with unit u and nilpotent t. The paper reveals the many facets of this notion for 2 2 × integral matrices.doi:10.18642/jantaa_7100122079 fatcat:yemptmhwfna2jhry6irnkeey3m