Let $R$ be a commutative ring; does every ideal in the $I$-adic completion of $R$


$$ \varprojlim\_i R/I^i $$


arise as the $I$-adic completion of some ideal inside of the original ring $R$? 



Ideals in the I-adic completion of a ring
No: take the ring $k[[t\_1, ..., t\_n, ...]]$ of power series of countably many variables, and let $I$ be the ideal generated by $r:=\sum t\_i^i$. Since there is no invertible $u$ such that $r=uP$, where $P$ is a polynomial, the ideal $I$ does not come from any ideal in the polynomial ring.


