Algebras in which non-scalar elements have small centralizers. (English) Zbl 1335.16012
Let \(A\) denote a unital algebra over a field \(F\); and for \(a\in A\), denote by \(C(a)\) the centralizer of \(a\). The paper characterizes \(A\) such that for each \(a\in A\setminus F\), \(C(a)=F[a]\); and it also characterizes finite-dimensional \(A\) over perfect fields \(F\) such that \(C(a)\) is commutative for all \(a\in A\setminus F\).
Reviewer: Howard E. Bell (St. Catharines)
MSC:
| 16K20 | Finite-dimensional division rings |
| 16P10 | Finite rings and finite-dimensional associative algebras |
| 16U80 | Generalizations of commutativity (associative rings and algebras) |
| 15A27 | Commutativity of matrices |
| 15A78 | Other algebras built from modules |
References:
| [1] | DOI: 10.1090/S0002-9947-1969-0236208-5 · doi:10.1090/S0002-9947-1969-0236208-5 |
| [2] | Amitsur SA, Pacific J. Math 137 pp 327– (1969) |
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| [4] | DOI: 10.1142/S100538671000060X · Zbl 1205.17009 · doi:10.1142/S100538671000060X |
| [5] | DOI: 10.1007/s00605-007-0505-1 · Zbl 1139.16017 · doi:10.1007/s00605-007-0505-1 |
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| [7] | DOI: 10.1112/blms/5.3.312 · Zbl 0267.16016 · doi:10.1112/blms/5.3.312 |
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