Nil-clean and unit-regular elements in certain subrings of ${\mathbb M}_2(\mathbb Z)$

Abstract

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Characterizations of special clean elements and applications

“…In [37] As also A = 3 1 0 0 is not strongly regular in R 2 (1 + A − AW −1 = 3 1 0 1 is not a unit, see for instance [32]), we get that a uniquely special clean element need not be strongly regular. Also the idempotent e related to the special clean decomposition (equivalently ē = a − u) is not central.…”

Unipotent and Unit-Regular Elements in Certain Subrings of M_2 (Z)

“…In [1], Andrica and Calugareanu found a counter example and gave a structure theorem which is nil-clean but not clean element in the matrix ring 𝑀 2 (ℤ). In [8] the authors considered this problem on the subring ( ℤ ℤ 𝑠 2 ℤ ℤ ) of 𝑅 ≔ 𝑀 2 (ℤ) instead of 𝑅 since the subring ( ℤ ℤ 𝑠 2 ℤ ℤ ) contains much less clean elements than 𝑀 2 (ℤ), a huge advantage. The authors of [8] gave also many counter-examples of unit-regular elements (an element in a ring is unit-regular if it is a product of an idempotent and a unit, and a ring is unit-regular if its every element is unit-regular) and nil-clean elements that are not clean in the ring ( ℤ ℤ 𝑠 2 ℤ ℤ…”

Jacobsons Lemma fails for nil-clean 2 × 2 integral matrices

“…Moreover, Jacobson's lemma holds for strongly nil-clean elements and fails for nil-clean elements. An example in a subring of M 2 (Z) was recently given in [5].…”