Hey!!

Based on the definition of $+_k$, I want to give an inductive definition for the multiplication $\cdot_k$ in $\mathbb{Z}_k$, such that for all $x\in \mathbb{N}_0$ and $y\in \mathbb{N}_0$ it holds that $$x\cdot_k y=(x\cdot y)\mod k$$

What is an inductive definition?

Do we write $x\cdot y=y+y+\ldots +y$ ($n$-times) and we apply each time the addition $+_k$ ?

I want to prove also by induction that for a $x\in \mathbb{N}_0$ it holds for all $y\in \mathbb{N}_0$ that $$(x\cdot y)\mod k=(x\mod k)\cdot_k (y\mod k)$$

We apply the induction on $y$ or not?