Conjugate idempotent formal matrices of order 2 over residue class rings

Let p be a prime, p > 1, m and n be integers, m > n > 0. In recent works [1-5] the following formal matrix rings were considered: 'Zlp"Z ZIp'Z) If a + p"Z b + p'Z ,Z Ip"Z ZIp"Z =[[ c + p"Z d + p"Z with multiplication defined so that for every A, A'e K we have f a + pmZ b + p"Z a' + pmZ b'+ p"Z c + p"Z d + p"Z j\ c+ p"Z d' + p"Z f aa+ p"-"bc' + pmZ ab'+ bd' + p"Z ca' + dc' + p"Z pm-"cb' + dd' + p"Z f a + pmZ b + p"Z у c + p" Z d + p" Z and only ifp divides (does not divide) a and d. In [1] it was shown that A is a nontrivial idempotent in K if and only if A has the form f 1 "n Л K = A · A' = It is known [2-5] that the matrix A = where b, c e 1-CT + pm Z b + p" Z c + p" Z CT + p" Z v+l Z, CT = YC, (pm-"bcf , k=1 a,b,c,d e Z K is nilpotent (invertible) if CT + pm Z b + p" Z У c + p" Z 1-CT + p" Z " -1 and Ci are Catalan numbers. For m - " .. 1 f 2i - 2 every i > 0 we define the ith Catalan number by 1 i у i -1 Д21 - 2)! so C1 = 1, i!(i -1)! C2 = 1, C3 = 2, C4 = 5, C5 = 14, etc. Let us call a non-trivial idempotent matrix with an invertible element in the upper left corner an idempotent matrix of type 1. An idempotent matrix of type 2 is a non-trivial idempotent matrix with an invertible element in the lower right corner. Definition 2.1. Idempotent elements e 1 and e2 of ring R are conjugate if there is an invertible element u e R such that e2 = ue1u“'. We have obtained the following results. Theorem 2.3. In the formal matrix ring K every idempotent matrix of type 1 is conjugate to the matrix К 1 I m r\ I "1 + p Z 0 + p Z 11 у0+p"Z 0 + p"Z conjugate to the matrix E22 = . Likewise, every idempotent matrix of type 2 is r\ I m r\ I " 0 + p Z 0 + p Z 0 + p" Z 1 + p" Z. Corollary 2.4. In the formal matrix ring K two idempotent matrices of different types are never conjugate. Corollary 2.6. In the formal matrix ring K any two idempotent matrices of the same type are conjugate.

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