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  • Author or Editor: W. Nicholson x
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A ringR is called a leftPP-ring if every principal left ideal is projective, equivalently if the left annihilatorl(a) is generated by an idempotent for alla∈R. These rings seem first to have been discussed by Hattori [2] and examples include (von Neumann) regular rings and domains (possibly noncommutative). In this note we give a new characterization of leftPP-rings, use that to give an elementary proof of a result of Xue [4] characterizing triangularPP-rings, and then determine when the ringT n (R) of upper triangular matrices overR is a leftPP-ring. Throughout the paper all rings have a unity and all modules are unitary.

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An operatorT:V?V on a real inner product space is called complement preserving if, wheneverU is aT-invariant subspace ofV the orthogonal complementU ? is alsoT-invariant. In this note we obtain some results on such operators.

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