Nil-clean elements which are not clean in certain subrings of M3(ℤ)

Abstract

Let R be a ring with identity. An element x ∈ R is said to be clean if x = u + e for some unit u and idempotent e in R. If x = u + e or x = u − e for some unit u and idempotent e in R, then x is said to be weakly clean. The element x is said to be nil-clean if it is the sum of an idempotent and a nilpotent element. The ring R is called clean (weakly clean, nil-clean) if all of its elements are clean (weakly clean, nil-clean, respectively). It is known that nil-clean rings are clean. However, a nil-clean element is not necessarily clean. We give more examples of this by showing the existence of elements which are nil-clean but not clean in a subring of M3(ℤ), the ring of 3 × 3 matrices over ℤ. As a consequence, we also obtain examples of weakly clean elements which are not clean.