A nil-clean 2 × 2 matrix over the integers which is not clean

Abstract

While any nil-clean ring is clean, the last eight years, it was not known whether nil-clean elements in a ring are clean. We give an example of nil-clean element in the matrix ring ℳ2(Z) which is not clean.

Nil-clean and strongly nil-clean rings

“…(3) It is similar to (2). P An element of a ring is called uniquely clean if it has only one clean decomposition in the ring.…”

“…Furthermore, the nil-cleanness of a matrix ring is tightly linked to the famous Köthe Conjecture (see Section 6). The reader is referred to the papers [10,5,16,2] for the background and current stage of the study of nil-clean and strongly nil-clean rings.…”

Characterization of $2\times 2$ nil-clean matrices over integral domains

“…where A, B, C, D, E and F are integers. We shall use the notations of [3] and follow the method of solution given there. Let D = B 2 − 4AC, g = gcd(B 2 − 4AC, 2AE − BD) and = 4ACF + BDE − AE 2 − CD 2 − F B 2 .…”

“…This question was answered in negative by Andrica Dorin and Calugareanu Grigore in [3]. They have given an example of a matrix in M 2 (Z Z) which is nil-clean but not clean.…”

Certain decompositions of matrices over Abelian rings

“…An element a in a ring is called weak idempotent if a or −a is an idempotent. An element in R is (weakly) nil clean provided that it is the sum of a (weak) idempotent and a nilpotent element [3], [5], [9], [10], and [12]. A ring R is (weakly) nil clean if every element in R is (weakly) nil clean.…”