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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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