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## Main Question or Discussion Point

For a commutative ring [tex]R[/tex] and an ideal [tex]I[/tex], is it true that [tex]I \oplus R/I \cong R[/tex] ? I know in some cases this is true, and I know it's true for finitely-generated Abelian groups, but is it true for any commutative ring?

In other words, we know that [tex]R/I[/tex] is isomorphic to some ideal in [tex]R[/tex], call this [tex]J[/tex]. It's clear that [tex]I \cap J = 0[/tex], but does [tex]I + J = R[/tex] ?

In other words, we know that [tex]R/I[/tex] is isomorphic to some ideal in [tex]R[/tex], call this [tex]J[/tex]. It's clear that [tex]I \cap J = 0[/tex], but does [tex]I + J = R[/tex] ?