Isomorphism between quotient of ring and principal idealsLet R be a principal ideal domain and suppose that u,u′∈R are such that R(u)=∼R(u′) as R-modules, where (u) denotes the ideal generated by u. Is it true that u=αu′, where α∈R is invertible?

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Kaitlynn Craig · Accepted Answer

Step 1If ︎R(u)≃︎R(u′)as R-modules, then they have the same annihilator (why?), so(u)=(u′)and thusu=αu′with α∈Rinvertible.In this answer R is an integral domain.

Isomorphism between quotient of ring and principal ideals Let

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