How to define a ring $R$ and two non-zero polynomials $a(x), b(x) \in R[x]$

Question

such that the degree of their product is NOT equal to the sum of their degrees?

Could someone please help with this question? Thank you.

Answer 1

In $\mathbb{Z}/4\mathbb{Z}[X]$, consider the polynomials $a=b=2X$. Each of them is of degree $1$ but their product is $4X^2=0X^2=0$.

Answer 2

For two polynomials, the product is given by: $$(a_nx^n + ,\dots,+a_0)(b_mx^n + , \dots,+ b_0) = a_nb_mx^{m+n} + \Big( \text{lower degree terms} \Big) $$

So the only way for the product of these polynomials to have a degree different than their sum is for $a_nb_m$ to be $0$.

One way to do this is to choose a ring that has zero-divisors. Examples have been given in other answers. Another way to do this is to simply let one of the polynomials be the $0$-polynomial