Let $R$ be a non-zero ring with identity. Is it possible for $R[x]$ to have only a finite number of maximal left ideals !? 



Maximal Ideals in $R[x]$
Lets try something like Euclid's proof of infiniteness of prime numbers. Let $f\_1 = x, f\_2 = xf\_1 + 1, f\_3=xf\_1f\_2+1, f\_4=xf\_1f\_2f\_3 + 1$ and so on. The left ideal generated by each $f\_i$ is proper. Thus each $f\_i$ is contained in a maximal left ideal. On the other hand for any $i < j$ we have $ f\_i | f\_j-1$ (note that $f\_i$'s are in the center of $R[x]$). This means that $f\_i$ and $f\_j$ can not be both in one maximal left ideal. Hence we have an infinite number of maximal left ideals. 


