Is there a way of finding all possible isomorphisms on the direct product of rings? I know that if the rings are of different size, direct product isomorphism induces automorphism on each rings, and if the rings are the same (isomorphically), then it also induces isomorphism in between the structures in addition to the automorphisms, but are these the only possibilities?
2026-03-26 16:06:05.1774541165
Direct Product of Rings and isomorphism
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In general not, it highly depends on what your rings in the product are. But you can reformulate the question in light of catogory theory: by universal property of products endomorphisms $\phi: A \times B \to A \times B$ are 1-1 to pairs $(\phi_A:A \times B \to A, \phi_B:A \times B \to B)$. Sometimes it helps when you have more informations about $A$ and $B$.