Strongly clean triangular matrix rings with endomorphisms

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Abstract: A ring

R

is strongly clean provided that every element in

R

is the sum of an idempotent and a unit that commutate. Let

T_{n} (R, s i g m a)

be the skew triangular matrix ring over a local ring

R

where

s i g m a

is an endomorphism of

R

. We show that

T_{2} (R, s i g m a)

is strongly clean if and only if for any

a i n 1 + J (R), b i n J (R)

,

l_{a} - r_{s i g m a (b)} : R o R

is surjective. Further,

T_{3} (R, s i g m a)

is strongly clean if

l_{a} - r_{s i g m a (b)}, l_{a} - r_{s i g m a^{2} (b)}

and

l_{b} - r_{s i g m a (a)}

are surjective for any

a i n U (R), b i n J (R)

. The necessary condition for

T_{3} (R, s i g m a)

to be strongly clean is also obtained.

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