For proper nonzero ideals J₁,J₂ of a nonzero commutative ring R, P:for every r₁,r₂∈R there is a unique r∈R with r−r₁∈J₁,r−r₂∈J₂. Q:J₁+J₂=R. Which implication holds?
P gives existence for r₁=0 and arbitrary r₂, writing r₂ as a sum of an element of J₁ and one of J₂, hence Q. Comaximality guarantees existence only modulo J₁∩J₂, not uniqueness as an actual ring element. In ℤ, J₁=2ℤ,J₂=3ℤ are comaximal but any solution can be shifted by 6. Thus **A**.
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