• Title/Summary/Keyword: right (left) quasicentral idempotent

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STRUCTURE OF IDEMPOTENTS IN POLYNOMIAL RINGS AND MATRIX RINGS

  • Juan Huang;Tai Keun Kwak;Yang Lee;Zhelin Piao
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.5
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    • pp.1321-1334
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    • 2023
  • An idempotent e of a ring R is called right (resp., left) semicentral if er = ere (resp., re = ere) for any r ∈ R, and an idempotent e of R∖{0, 1} will be called right (resp., left) quasicentral provided that for any r ∈ R, there exists an idempotent f = f(e, r) ∈ R∖{0, 1} such that er = erf (resp., re = fre). We show the whole shapes of idempotents and right (left) semicentral idempotents of upper triangular matrix rings and polynomial rings. We next prove that every nontrivial idempotent of the n by n full matrix ring over a principal ideal domain is right and left quasicentral and, applying this result, we can find many right (left) quasicentral idempotents but not right (left) semicentral.