Describe the ring $\mathbb{Z}[x]/(x^2)$. 
Describe the ring $\mathbb{Z}[x]/(x^2)$.

Attempt: $\mathbb{Z}[x]/(x^2)$ can be thought as the linear polynomials with integer coefficients.
So $\mathbb{Z}[x]/(x^2) = \{a + bx + (x^2) : a,b \in \mathbb{Z}\}$.
What are other things I could describe?
Thank you!

Abstractly, it's the ring $\mathbb Z[u]$ in which $u^2=0$, and so its elements are of the form $a+bu$, with $a,b\in\mathbb Z$.
In this sense, it's like $\mathbb Z[i]$ in which $i^2=-1$. But they are very different rings, of course.
If you want to understand $\mathbb Z[u]$ a little better, you may want to find its units, its zero divisors, its nilpotent elements, its idempotents, its ideals, etc.
In particular, you'll find that there are no non-trivial idempotents in $\mathbb Z[u]$ and so it cannot be expressed as the product of two rings.
