Abstract
In the paper we prove the double centralizer theorem for semiprime algebras. To be precise, let R be a closed semiprime algebra over its extended centroid F, and let A be a closed semiprime subalgebra of R, which is a finitely generated module over F. Then C R (A) is also a closed semiprime algebra and C R (C R (A)) = A. In addition, if C R (A) satisfies a polynomial identity, then so does the whole ring R. Here, for a subset T of R, we write C R (T): = {x ∈ R|xt = tx ∀ t ∈ T}, the centralizer of T in R.
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Members of Mathematics Division, National Center for Theoretical Sciences (Taipei Office).
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Chuang, CL., Lee, TK. The Double Centralizer Theorem for Semiprime Algebras. Algebr Represent Theor 17, 1277–1288 (2014). https://doi.org/10.1007/s10468-013-9447-6
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DOI: https://doi.org/10.1007/s10468-013-9447-6
Keywords
- Double centralizer theorem
- Closed (semi)prime algebra
- PI-ring
- Quasi-Frobenius algebra
- Extended centroid
- Orthogonal completion