# Prove that every nonzero quotient ring element is zero divisor.

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.

• Can you prove the analogous result in $\mathbf Z/m\mathbf Z$? It is the same idea. (The notation $T$ for a field is strange. Why not $F$? Or $K$?) – KCd May 15 at 14:29
• I use F notation a lot in my solutions, so it is uncomfortable for me to use it specifically for that purpose, however haven't heard of K-notation, will use it then next :) – 9cloudalpha May 15 at 14:32
• Writing about polynomials $f(x)$ in $F[x]$ looks okay. The letter $K$ is widely used for fields since it is the first letter of the German word for field (in algebra). – KCd May 15 at 14:38
• Ok, got it, thanks for the info! – 9cloudalpha May 15 at 14:39

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 $$a_0a=1$$.
• @9cloudalpha: Why not? Did you not have linear algebra? If you know what a vector space is, this is pretty easy to show. The dimension is the degree of $g$ (provided it is positive). – tomasz May 15 at 14:37
• Yeah, I know what vector space is, but that fact seems sound non-obvious to me, we have never observed $T[x]/(g)$ as vector space over T, actually – 9cloudalpha May 15 at 14:40
• @9cloudalpha: it's quite straightforward. Any ring containing $T$ is a vector space over $T$. – tomasz May 15 at 15:50