Given the following definitions for $\mathbb{Z}[x] /\left(x^{n}-1\right)$:
$$ a \cdot b \equiv \sum_{i=0}^{n-1} \sum_{j=i+1}^{n-1} a_{i} \cdot b_{j} \cdot x^{i+j}+\sum_{j=1}^{n-1} \sum_{i=n-j}^{n-1} a_{i} \cdot b_{j} \cdot x^{i+j-n}\left(\bmod x^{n}-1\right) $$ Similarly, for $\mathbb{Z}[x] /\left(x^{n}+1\right)$ the multiplication is defined as $$ a \cdot b \equiv \sum_{i=0}^{n-1} \sum_{j=i+1}^{n-1} a_{i} \cdot b_{j} \cdot x^{i+j}-\sum_{j=1}^{n-1} \sum_{i=n-j}^{n-1} a_{i} \cdot b_{j} \cdot x^{i+j-n}\left(\bmod x^{n}+1\right) $$
An examples details: Let $a(x) = x^{2} + 2x + 3$ and $b(x) = x^{2} + x$
The following examples are taken from a published work. Assuming the author used the above formulas to compute the final sums correctly:
Example 1 says that: In $\mathbb{Z}[x]/(x^{3} - 1)$ resulting sums from first formula are given as $(5x^{2} + 3x) + (x + 3) = 5x^{2} + 4x + 6$.
Question 1: How is 6 obtain in final answer? should it not be $5x^{2} + 4x + 3$? because $\mathbb{Z}[x]$ means we are working with polynomials in $x$ whose coefficients are defined over $\mathbb{Z}$, the set of all integers.
Example 2: In $\mathbb{Z}[x]/(x^{3} + 1)$ resulting sums from second formula are $(5x^{2} + 3x) - (x + 3) = 5x^{2} + 2x$.
Question 2: Similarly, should the resulting answer not be $5x^{2} + 2x - 3$ since there is restriction on the coefficients (e.g., we are not working in $\mathbb{Z}_q$ for some specified $q$).
