Answer to Question #137501 in Abstract Algebra for Ram

Question

consider the ideal I=12Z of Z. Find a proper ideal J of Z such that I+J=Z

Answer 1

Let us consider the ideal $J=11\mathbb Z$ of $\mathbb Z$. Then $1=12+(-11)\in 12\mathbb Z +11\mathbb Z=I+J$. Since $I+J$ is an ideal in $\mathbb Z$, $a=1\cdot a\in I+J$ for any $a\in\mathbb Z$. Therefore, $\mathbb Z\subset I+J.$ Taking into account that $I+J\subset \mathbb Z$, we conclude that $I+J=\mathbb Z.$