Equivalence of diagonal matrices over local rings

It is proved that two diagonal matrices diag(a1,…,an) and diag(b1,…,bn) over a local ring R are equivalent if and only if there are two permutations σ,τ of {1,2,…,n} such that [R/aiR]l=[R/bσ(i)R]l and [R/aiR]e=[R/bτ(i)R]e for every i=1,2,…,n. Here [R/aR]e denotes the epigeny class of R/aR, and [R/aR]l denotes the lower part of R/aR. In some particular cases, like for instance in the case of R local commutative, diag(a1,…,an) is equivalent to diag(b1,…,bn) if and only if there is a permutation σ of {1,2,…,n} with aiR=bσ(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.