Isomorphism between quotient of ring and principal ideals Let

Rosa Townsend 2022-04-13 Answered
Isomorphism between quotient of ring and principal ideals
Let R be a principal ideal domain and suppose that u,uR 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 αR is invertible?
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Answers (1)

Kaitlynn Craig
Answered 2022-04-14 Author has 13 answers
Step 1
If R(u)R(u)
as R-modules, then they have the same annihilator (why?), so
(u)=(u)
and thus
u=αu
with αR
invertible.
In this answer R is an integral domain.
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