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I would be grateful if anyone could give me a reference regarding the following question.

Suppose that $A$ and $B$ are two unital prime algebras over a field $F$ whose center consists of scalar multiples of identity (so called central algebras). Assume that both $A$ and $B$ have a unique non-trivial two-sided ideal, denote it respectively by $J_A$ and $J_B$. I'm interested about the ideal structure of a tensor product $A \otimes B$ (over $F$). The obvious non-trivial ideals of $A \otimes B$ are $J_A \otimes J_B$, $A \otimes J_B$, $J_A \otimes B$ and $J_A\otimes B + A \otimes J_B$. Are there some conditions on $A$ and $B$ that guarantee these are the only non-trivial ideals of $A \otimes B$?

I'm particularly interested in a case when $A=B$ is the algebra of $F$-linear maps on some vector space over $F$ of infinite countable dimension. Then the only non-trivial ideal of $A$ is the ideal $J_A$ of finite rank maps. Are all ideals of $A \otimes A$ of the above form?

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Well, i do not know the answer in general but since you are asking for a reference and if

there are some conditions on $A$ and $B$ that guarantee these are the only non-trivial ideals of $A \otimes B$?

There are some related results, for example:
Let $B$ an arbitrary algebra over a field.

Every ideal of the algebra $A\otimes B$, where $A$ is a central simple algebra, is of the form $A\otimes I$, where $I$ is an ideal of the algebra $B$.

This is theorem 4.3.2, p. 74, from the book of Drozd-Kirichenko.
(although i suspect this may not be of much use for the particular case you are interested in, as explained in the last paragraph of the OP).

Maybe this reference might also be of some interest to you.

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  • $\begingroup$ Thanks, but I'm aware of all this. I'm interested if something can be said abut the ideal structure $A \otimes B$ when both central algebras $A$ and $B$ have a unique non-trivial two-sided ideal. $\endgroup$ – Ilja Apr 4 '19 at 8:28
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I think it is true for your example.

By a similar proof as mentioned by Konstantinos Kanakoglou, we have the following fact.

  • Let $I$ be an ideal of $A \otimes B$. If $I \nsubseteq J_A \otimes J_B$, then $I = J_A \otimes B, A \otimes J_B$ or $J_A \otimes B + A \otimes J_B$.

Now suppose that $A=B$ is the algebra of $F$-linear maps on some vector space over $F$ of infinite countable dimension. For all $0 \neq \sum x_i\otimes y_i \in J_A\otimes J_A$, $x_i,y_i(\in J_A)$ can be seen as endmorphisms of a finite dimensional vector space. Hence $\exists 0 \neq x,y \in J_A$, such that $x \otimes y \in $ the ideal generated by $\sum x_i\otimes y_i$. The ideal generated by $x \otimes y$ is just $J_A \otimes J_A$, then we have

  • If $I$ is a subideal of $J_A \otimes J_A$, then $I = 0$ or $J_A \otimes J_A$.

So $A\otimes A$ has only four nonzero ideals.

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  • $\begingroup$ In the mean time I proved this is indeed true for $\mathrm{End}_F(V) \otimes \mathrm{End}_F(V)$, where $V$ is a vect. space of infinite countable dimension over a filed $F$. But if $A$ and $B$ are as in my original post and $I$ an ideal of $A \otimes B$ such that $I$ is not contained in $J_A \otimes J_B$, why is $I$ is necessarily of the form $J_A \otimes B$, $A \otimes J_B$ or $J_A \otimes B+ A \otimes J_B$? BTW, if both $A$ and $B$ are centrally closed then $J_A \otimes J_B$ is indeed the smallest proper ideal of $A \otimes B$ by the main result of Nicholson-Watters' paper from 1983 (PAMS). $\endgroup$ – Ilja Apr 13 '19 at 8:00
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    $\begingroup$ Thanks. Also like the proof of (1) $\Rightarrow$ (2) in Nicholson-Watters' main result. If $I$ is not contained in $J_A \otimes J_B$, we can assume that $r_1 = 1$ or $s_1 = 1$. So $0 \neq 1 \otimes s_1 \in I$ or $0 \neq r_1 \otimes 1 \in I$. Hence $ J_A \otimes B \subset I$ or $ A \otimes J_B \subset I$. So we can assume that $A$ or $B$ is a simple algebra. $\endgroup$ – rpz Apr 13 '19 at 10:43
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I don't think the solution to this problem is as simple as rpz says. In fact, I wrote a short paper about this so, in case if anyone is interested, please check Thm. 3.9 of https://web.math.pmf.unizg.hr/~ilja/preprints/ISTPNSA.pdf

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