First, $A\times\mathbb{Z}$ is not a good notation for the ring (known as the Dorroh extension of $A$), because $A\times\mathbb{Z}$ has a natural ring structure which is not the one we are using here. So it's best to call it something else. It's common to use $A^1$.
Second: There is a canonical embedding $\varphi\colon A\to A^1$ given by $\varphi(a)=(a,0)$. Under this embedding, you can verify rather easily that $\varphi(I) = \{(x,0)\mid x\in I\}$ is a left (resp. right, two-sided) ideal of $A^1$ whenever $I$ is a left (resp. right, two-sided) ideal of $A$.
If $J$ is a left (resp. right, two-sided) ideal of $A^1$, then both the intersection with $\varphi(A)$ and the projection onto $A$ are left (resp. right, two-sided) ideals of $A$.