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Consider the following conditions on two proper non zero ideals $J_{1}$ and $J_{2}$ of non zero commutative ring R

P: for any $r_{1}, r_{2}\in R$ there exists a unique $r \ in R$ such that $r-r_{1}\in J_{1} $ and $r- r_{2}\in J_{2}$.

Q:$J_{1} +J_{2} =R$

Then which of the following is true?

A) P implies Q and Q implies P

B) P implies Q and Q does not imply Q

C) P does not imply Q and Q implies P

D) neither P implies Q nor Q implies P.

By definition of ideal $r-r_{1}\in J_{1} $ and $r- r_{2}\in J_{2}$. But what uniqneness of r has to do here. Which of the given option is true and how? A single correct option question of ring theory.

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  • $\begingroup$ What Uniqueness has to do here in P? $\endgroup$
    – Sejy
    Oct 6, 2022 at 5:25
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    $\begingroup$ By including "unique" in the statement of $P$, it makes it possible to show that $Q$ does NOT imply $P$. For if it happens that $0\neq x\in J_1\cap J_2$, then it's impossible for the $r$ in $P$ to be unique. $\endgroup$
    – rschwieb
    Oct 6, 2022 at 13:51

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