Pseudorandomness of ring learning with errors
Let $R$ be the ring $\mathbb{Z}_p[X]/(X^n+1)$, where $n$ is a power of 2.  The Ring-LWE assumption says that for any randomly chosen, fixed $s\in R$, the distribution of $$((a_1,a_1s+e_1),\ldots,(a_k,a_ks+e_k))$$ is indistinguishable from the distribution $$((a_1,u_1),\ldots,(a_k,u_k)),$$ where $a_i,u_i$ are uniformly random in $R$ and $e_i$ are chosen from some distribution with small coefficients.

Of course, if one $only$ outputs $((a_1s+e_1),\ldots,(a_ks+e_k))$ without also revealing the $a_i$, then this distribution is trivially uniformly random (assuming that $s$ is invertible in $R$, but this is a minor issue) because the $a_i$ are uniformly random.  This would be a pretty useless property.  

The interesting part in Ring-LWE is that even when the $a_i$ are revealed, the distribution still looks uniform.  Basically, the small additional entropy of the $e_i$ allows one to create a new uniform-looking element with larger entropy.