In Artin Algebra, page 340, Example 11.5.7 it said that if we quotient out $\delta^2-3$ in the field $\mathbb F_{11}$ to add two abstract square roots of 3, since we are not told that these square roots are $5$ and $-5$, we know that $\delta-5$ and $\delta+5$ are not zero. But since $\delta^2-3=0$, $(\delta-5)(\delta+5)$ is zero. And it cannot happen in a field.
So does it mean that in a ring but not a field we can get a $0$ from 2 non zero element? If so, could someone give me a such example?
Thanks in advance!
This is the concept of a zero divisor: these are non-zero elements $a$ for which there exist a non-zero $b$ such that $ab = 0$. These do not exist in fields since fields are a subset of those special rings known as integral domains which have no zero divisors.
As you remarked, the quotient ring $\mathbb{Z}/m\mathbb{Z}$ where $m$ is not prime is a classic example. Any element $g$ with $(g, m) \neq 1$ is a zero divisor. For example, $4$ in the quotient ring $\mathbb{Z}/14\mathbb{Z}$ is a zero divisor since $4 * 7 = 0$.