If $R/I \times R/J$ is isomorphic to $R/(I\cap J)$ as $ R $-modules, then $I + J = R$.
32343
Converse to Chinese Remainder Theorem
154754
Is $O_n$ isomorphic to $SO_n \times \{\pm I\}$?
145722
Prove there is a bijection between the set of $R[x]$-module structures on $M$ and $\mathrm{End}_{R-\mathsf{Mod}}(M)$.
124646
Understanding the ideal $IJ$ in $R$
120183
Prove that $S_3\times\mathbb{Z}_2$ is isomorphic to $D_6$
108044
Isomorphic extensions of $\mathbb Z_3$ by $\mathbb Z_3 \times \mathbb Z_3$ which are not equivalent
6896
If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$
157840
Is it true that $(R\times S)[G]\cong R[G]\times S[G]$?
157236
Tensor product of modules.
75641
Is there a (f.g., free) module isomorphic to a quotient of itself?
97608
Showing a Ring of endomorphisms is isomorphic to a Ring
59995
Isomorphism between $\mathbb{C}^* $ under multiplication with $\mathbb{C}$ under addition.
81197
Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?
137011
showing $\psi: R\to \mathbb C$ is ring isomorphism.
70658
Isomorphism between quotient modules
84869
Are the groups $\mathbb{C}$ and $\mathbb{R}$ isomorphic?
86034
Is it true that for algebraic sets $V,W$ we have $I(V \times W ) =I(V) + I(W)$?