Strongly Nil-*-Clean Rings

Publication date: 22 November 2012

Abstract: A *-ring

R

is called a strongly nil-*-clean ring if every element of

R

is the sum of a projection and a nilpotent element that commute with each other. In this article, we show that

R

is a strongly nil-*-clean ring if and only if every idempotent in

R

is a projection,

R

is periodic, and

R / J (R)

is Boolean. For any commutative *-ring

R

, we prove that the algebraic extension

R [i]

where

i^{2} = m u i + e t a

for some

m u, e t a i n R

is strongly nil-*-clean if and only if

R

is strongly nil-*-clean and

m u e t a

is nilpotent. The relationships between Boolean *-rings and strongly nil-*-clean rings are also obtained.

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