Answer by user26857 for Direct sum of non-zero ideals over an integral domain
If $R$ is a commutative ring, and $L\subset R$ is an ideal such that $R\oplus L\simeq R^2$ then $L\simeq R$. We have $\bigwedge^2(R\oplus L)\simeq\bigwedge^2R^2$, so $L\simeq R$. (Here I've used that...
View ArticleAnswer by user26857 for Direct sum of non-zero ideals over an integral domain
In this answer you can find two principal ideals $I,J$ such that $I\cap J$ is not finitely generated. Then you can't have $R\oplus (I\cap J)\simeq I\oplus J$ since then $R\oplus (I\cap J)\simeq R\oplus...
View ArticleDirect sum of non-zero ideals over an integral domain
Let $R$ be an integral domain. Let $I$ and $J$ be non-zero ideals of $R$. Is this statement always true: $$R\oplus(I\cap J)\cong I\oplus J\ ?$$I regarded the short exact sequence $0\to I\cap J\to...
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