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I am having trouble coming up with the explicit isomorphism. For I(R/J) I know any element can be expressed as i(r + J) = ir + J by definition of the action of R on R/J.

As for (I + J)/J, any element can be expressed as i + j + J = i + J so i was thinking of mapping i + J to ir + J but the problem is that this map doesn't seem to be 1 - 1.

But I am having trouble coming up with any other sensible maps besides this one. Can someone offer a hint on how to proceed?

Thanks!