Equivalence of diagonal matrices over local rings

Abstract

It is proved that two diagonal matrices diag ( a 1 , … , a n ) and diag ( b 1 , … , b n ) over a local ring R are equivalent if and only if there are two permutations σ , τ of { 1 , 2 , … , n } such that [ R / a i R ] l = [ R / b σ ( i ) R ] l and [ R / a i R ] e = [ R / b τ ( i ) R ] e for every i = 1 , 2 , … , n . Here [ R / a R ] e denotes the epigeny class of R / a R , and [ R / a R ] l denotes the lower part of R / a R . In some particular cases, like for instance in the case of R local commutative, diag ( a 1 , … , a n ) is equivalent to diag ( b 1 , … , b n ) if and only if there is a permutation σ of { 1 , 2 , … , n } with a i R = b σ ( i ) R for every i = 1 , … , n . These results are obtained studying the direct-sum decompositions of finite direct sums of cyclically presented modules over local rings. The theory of these decompositions turns out to be incredibly similar to the theory of direct-sum decompositions of finite direct sums of uniserial modules over arbitrary rings.