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!
 A: 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.
A: It's a vague question, so the answer might also turn out vague.
As you have already done, describe the elements of the ring. What else? You could describe the addition and the multiplication in the ring, and you could figure out which elements have a multiplicative inverse.
A: 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.
A: Is $\mathbb{Z}[x]/(x^2)$ a field? Is it a domain? Is it a reduced ring?
What can you deduce from the ideal $(x^2)$?
