Is $\mathbb{Z}[x,y]/((x+1, y+1) \cong \mathbb{Z}$ ?
My attempt : i think yes
consider the map $\phi : \mathbb{Z}[x,y] \to \mathbb{Z}$ defined by $\phi(f(x,y)) = f(-1,-1)$. $\phi$ is a ring homomorphism with $\ker(\phi) = \{ f(x,y) \in \mathbb{Z}[x,y] : f(-1,-1) = 0 \}$. We will show that the kernel is the principal ideal $(x+1,y+1)$. This will imply, from the first isomorphism theorem, that $\operatorname{im}(\phi) \cong \mathbb{Z}[x,y]/((x+1, y+1)$, which gives an explicit description of the quotient.
Is its true ?
Best Answer
The factor ring ${\Bbb Z}[x,y]/\langle x+1,y+1\rangle$ has the monomial standard basis $\{1\}$ and so is isomorphic to $\Bbb Z$ as a $\Bbb Z$-module, but also a ring.