SEMICENTRAL IDEMPOTENTS IN A RING
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Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R and Sℓ(R) (resp. Sr(R)) be the set of all left (resp. right) semicentral idempotents in R. In this paper, the following are investigated: (1) e ∈ Sℓ(R) (resp. e ∈ Sr(R)) if and only if re = ere (resp. er = ere) for all nilpotent elements r ∈ R if and only if fe ∈ I(R) (resp. ef ∈ I(R)) for all f ∈ I(R) if and only if fe = efe (resp. ef = efe) for all f ∈ I(R) if and only if fe = efe (resp. ef = efe) for all f ∈ I(R) which are isomorphic to e if and only if (fe)ⁿ= (efe)ⁿ (resp. (ef)ⁿ= (efe)ⁿ) for all f ∈ I(R) which are isomorphic to e where n is some positive integer; (2) For a ring R having a complete set of centrally primitive idempotents, every nonzero left (resp. right) semicentral idempotent is a finite sum of orthogonal left (resp. right) semicentral primitive idempotents, and eRe has also a complete set of primitive idempotents for any 0 ≠ e ∈ Sℓ(R) (resp. 0 ≠ e ∈ Sr(R)).
Abstract
Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R and Sℓ(R) (resp. Sr(R)) be the set of all left (resp. right) semicentral idempotents in R. In this paper, the following are investigated: (1) e ∈ Sℓ(R) (resp. e ∈ Sr(R)) if and only if re = ere (resp. er = ere) for all nilpotent elements r ∈ R if and only if fe ∈ I(R) (resp. ef ∈ I(R)) for all f ∈ I(R) if and only if fe = efe (resp. ef = efe) for all f ∈ I(R) if and only if fe = efe (resp. ef = efe) for all f ∈ I(R) which are isomorphic to e if and only if (fe)ⁿ= (efe)ⁿ (resp. (ef)ⁿ= (efe)ⁿ) for all f ∈ I(R) which are isomorphic to e where n is some positive integer; (2) For a ring R having a complete set of centrally primitive idempotents, every nonzero left (resp. right) semicentral idempotent is a finite sum of orthogonal left (resp. right) semicentral primitive idempotents, and eRe has also a complete set of primitive idempotents for any 0 ≠ e ∈ Sℓ(R) (resp. 0 ≠ e ∈ Sr(R)).
참고문헌 (5)
- G. F. Birkenmeier, "Triangular matrix repre-sentations", J. Algebra, 230(2), pp.558~595 (2000)
- G. Calaugareanu, "Rings with lattices of idempotents", Comm. Algebra, 38(3), pp.1050~1056 (2010)
- H. K. Grover, "Rings with multiplicative sets of primitive idempotents", Comm. Algebra, 37(8), pp.2583~2590 (2009)
- J. Han, "Additive set of idempotents in rings", Comm. Algebra, 40(9), pp.3551~3557 (2012)
- T. Y. Lam, "A First Course in Noncommutative Rings" (1991), Springer-Verlag, New York, Inc.