Timeline for Ideal structure of a tensor product of certain algebras

3 events

when toggle format	what		by	license	comment
Apr 13, 2019 at 10:43	comment	added	rpz		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.
Apr 13, 2019 at 8:00	comment	added	Ilja		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).
Apr 11, 2019 at 9:57	history	answered	rpz	CC BY-SA 4.0