A ring homomorphism is a function between two rings that preserves the ring structure. This means that it maps the ring elements to their corresponding elements in the other ring, and also respects the operations of addition and multiplication.
In this case, the ring homomorphism maps elements of the ring $\mathbb{Z}[x]/(x^3-x)$ to elements in the ring $\mathbb{Z}$. It does this by taking the coefficients of the polynomial in $\mathbb{Z}[x]$ and evaluating them in $\mathbb{Z}$. The ideal $(x^3-x)$ is used to reduce the polynomial to a degree 2 or lower, as the ring $\mathbb{Z}$ only consists of constant and linear polynomials.
The ideal $(x^3-x)$ is used to ensure that the ring homomorphism is well-defined. It acts as a "kernel" for the homomorphism, meaning that any element in the ideal will be mapped to 0 in $\mathbb{Z}$, thus preserving the structure of the ring.
This ring homomorphism is related to the concept of quotient rings, as the ring $\mathbb{Z}[x]/(x^3-x)$ is a quotient of the polynomial ring $\mathbb{Z}[x]$. It is also related to the concept of isomorphism, as the ring $\mathbb{Z}[x]/(x^3-x)$ is isomorphic to the ring $\mathbb{Z}^2$.
Ring homomorphism has many applications in mathematics, including abstract algebra, number theory, and algebraic geometry. It can be used to study the structure of rings and their properties, and it also plays a role in solving equations and proving theorems in these areas of mathematics.