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!

This is actually a very interesting ring:
You can consider 'x' an "infinitesimal" in a sense.
Let $A = a + x a'$ and $B = a + x a'$ then we have


*

*$A + B = (a+b) + x(a'+b')$

*$AB = (ab) + x(a'b + ab')$


The first component is the normal operation, but the second component follows the same formulas as differentiation does!
This has applications in automatically differentiating functions in programming.
