Let $F$ be a field. Let $A$ be a simple (associative unital) $F$-algebra with center reduced to $F$. Let $B$ be a $F$-subalgebra of $A$; assume that $A$ is can be generated as left $B$-module by $n$ elements.
Does it follow that $\dim_F(Z(B))\le n$?
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1$\begingroup$ What exactly do you mean by $[Z(B):F]$ here? $\endgroup$– user1688Jan 10, 2016 at 9:27
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$\begingroup$ $Z(B)$ is centralizer of $B$ in $A$.and $[Z(B):A]$ is index of $Z(B)$ in $F$ $\endgroup$– tomJan 10, 2016 at 9:36
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$\begingroup$ $F$ is a submodule of $Z(B)$. $\endgroup$– tomJan 10, 2016 at 9:44
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$\begingroup$ yes exactly.I mean that $\endgroup$– tomJan 10, 2016 at 9:48
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1$\begingroup$ So $[Z(B):F]$ is the cardinal of $Z(B)/F$? what is $F$? If $F$ is an infinite field, then this index is either 1 or $\infty$. If you mean something else (e.g., you mean the codimension assuming $F$ is a field), please edit to make your question understandable. $\endgroup$– YCorJan 10, 2016 at 15:51
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