Equivalence of Diagonal Matrices over Local Rings

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 l[R/aiR]=l[R/bσ(i)R] and e[R/aiR]=e[R/bτ(i)R] for every i=1,2,…,n. Here e[R/aR] denotes the epigeny class of R/aR, and l[R/aR] denotes the lower part of R/aR. 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_iR=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.