Help calculating ideal generated by the union of two ideals

Question

I know this might seem a very elementary question, but i am really having troubles understanding the following :

let $R=\mathbb{Z}[X,Y]$, in my notes it is said that $I = ((X,Y,7) \cup (X)) = (X,Y,7)$ while $I' = ((X,Y,7) \cup (X+1,Y)) = (1)$. Can anyone explain why it is true? i am lacking some knowledge in Ring theory and i'm not able to see it.

Answer 1

For $I$, $X\in (X,Y,7)$ so we get nothing new by adding $(X)$...

Secondly an ideal is in particular an additive group. Since we have $X$ and $X+1$ in $I'$, we also have $X+1-X=1$. But once an ideal contains $1$ it is the whole ring.

Answer 2

The first equality results from the fact that $\;(X)\subset (X,Y,7)$.

Th second equality comes from the Bézout's relation: $$1\cdot(X+1)-1\cdot X=1,$$ hence the ideal generated by the union is the whole ring in this case.