Say I have $R= \mathbb{Z}[x]$ and $A = \{p_0+p_1x+p_2x^2+\cdots+p_nx^n \mid n\geqslant0, p_i\in\mathbb{Z}, p_0, p_1 \text{ even}\}$. Define $K=R/A$. How would I characterize the elements of $K$?
What I tried:
Since the only difference between the sequences of R and A is on $p_0$ and $p_1x$ (A has the extra condition that $p_0$ and $p_1$ need to be even), I concluded that in $R/A$ all that was left were the polynomials of the form:
p(x)=($p_0+p_1x$) where all values of $p_0$ and $p_1$ are in modulo (even numbers).
Looking at the positive integers that would leave me with 0 and 1 as possible values for $p_0$ and $p_1$ giving me the polynomials:
$0+0x$, $0+x$, $1+0x$, $1+x$
What I am wondering:
What about the negative integers?
Surely $-1 \mod (2) = 1$
but what about $-1 \mod (even integers)$? (-2 is still an even integer) and
$-1 \mod(-2) = -1$
From what I have concluded and had confirmed by a TA, is that the polynomials:
$-1-x$, $0-x$ and $-1+0x$
are also part of the elements of $K$ This stems from my earlier suspicion that indeed $mod(even integers)$ also includes $mod(-2)$.
(if anyone would like to dispute this conclusion I will leave it here for a day before I accept it as the answer)