M.F.M. Al Habibi, S. Irawati, H. Susanto, A.Y.M. Chin
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 M 3(), the ring of 3 x 3 matrices over . As a consequence, we also obtain examples of weakly clean elements which are not clean. © Published under licence by IOP Publishing Ltd.
Department of Mathematics, Universitas Negeri Malang, Jalan Semarang 5, Malang, 65145, Indonesia; Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur, 50603, Malaysia
