For the given field T and $g \in T[x]$ - polynomial of positive degree, prove that every nonzero and non-invertible quotient ring $T[x]/(g)$ element is indeed zero divisor.
This task was explained us during our zoom class, however I haven't really got it, but it seems to be fundamental one, so I need to realize its solution fully. Can you think of the most simple solution that would be really easy to undestand? Would be grateful for any help.
Not sure what is the easiest solution, but here's one I like.
Since $g$ is of positive degree, $R=T[x]/(g)$ is a finite-dimensional vector space over $T$. Let $a_0\in R$ be an element which is not a zero divisor. Then the map $R\to R$ given by the formula $a\mapsto a_0a$ is injective and $T$-linear, and thus it is surjective (since $R$ is a finite-dimensional vector space over $T$). In particular, there is some $a$ such that $a_0a=1$.