For non-commutative rings, we have this generalization of the Chinese remainder theorem (CRT). I wonder if there is another statement involving only left or right ideals; do you know any?
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$\begingroup$ It is easy to see that if $\{I_i:i<k\}$ are left ideals of a ring $R$, and $I=\bigcap_{i<k}I_k$, then $x+I\mapsto(x+I_0,\dots,x+I_{k-1})$ is an injective $R$-module homomorphism $R/I\to\bigoplus_{i<k}R/I_i$, which is surjective (i.e., an isomorphism) if and only if $I_i+\bigcap_{j\ne i}I_j=R$ for each $i<k$. I do not know if the latter condition can be simplified; in the case of two-sided ideals, it is equivalent to $\{I_i:i<k\}$ being pairwise coprime (i.e., $I_i+I_j=R$ for $i\ne j$), but I do not think this suffices for one-sided ideals. $\endgroup$– Emil JeřábekCommented Apr 25, 2023 at 12:07
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