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Semiprime algebras with finiteness conditions. (English) Zbl 1045.16008
The authors prove several appealing results on finite dimensionality in a semiprime algebra \(R\) over a field \(F\). The main theorem is that if \(a,b\in R\) and if \(\dim_FaIb\) is finite for some ideal \(I\) of \(R\), then \(\dim_FbIa=\dim_FaIb\). This result holds either with finite replacing finite dimensional or with \(aI\) replacing \(aIb\). If \(T\) is a right ideal of \(R\) and has zero left annihilator then for \(a\in R\), \(aT\) is finite dimensional over \(F\) (or is finite) forces \(aT=aR\). A final application shows that when every \(F\)-subalgebra of \(R\) with trivial multiplication is finite dimensional then \(R\) is a direct sum of a finite dimensional semiprime \(F\)-algebra and a reduced \(F\)-algebra.

MSC:
16N60 Prime and semiprime associative rings
16D25 Ideals in associative algebras
16P10 Finite rings and finite-dimensional associative algebras
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