EXTENSIONS OF CLEAN RINGS

By Juncheol Han, W. K. Nicholson. 2001

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On Strongly Clean Matrix and Triangular Matrix Rings (2006, background): “By Han and Nicholson ( 2001 ), a ring R is clean iff the matrix ring 𝕄 n ( R ) is clean for all n ≥ 1 iff the upper triangular matrix ring 𝕋 n ( R ) is clean for all n ≥ 1.”
Strongly clean matrix rings over commutative local rings (2008, background): “Prior to this, in [13], Han and Nicholson proved that if e ∈ R is idempotent, then R is a clean ring provided eRe and (1−e)R(1−e) are clean rings; a consequence of this is that if M1 and M2 are clean modules then M1 ⊕ M2 is clean.”
Strongly clean triangular matrix rings over local rings (2007, background): “Nicholson proved that strongly π -regular rings are strongly clean.”
Rings in which every element is either a sum or a difference of a nilpotent and an idempotent (2014, background): “When is Mn(R) weakly nil-clean? By [12], matrix rings over a clean ring are again clean as well as by [15, Proposition 2.”
Commutative weakly nil clean unital rings (2015, background): “Over the last ten to fifteen years there has been an explosion of interest in this class of rings as well as the many generalizations and variations (see [14] too).”