Adapting LWE Trapdoors for Ring-LWE
Instead of plain matrix, a matrix similar this (but bigger): 

+a -h -g -f -e -d -c -b
+b +a -h -g -f -e -d -c
+c +b +a -h -g -f -e -d
+d +c +b +a -h -g -f -e
+e +d +c +b +a -h -g -f
+f +e +d +c +b +a -h -g
+g +f +e +d +c +b +a -h
+h +g +f +e +d +c +b +a


which one can deduce it's equivalant in addition and multiplication to a polynomial reduced by $X^8+1$. And here, the polynomial is the "Ring". 

And instead of plain matrix of $n$ x $m$, we use a $1$ x $2$ matrix of polynomials. 

Note: the arithmetical difference between matrix-represented polynomial rings and plain matrices, is that the former is commutative in multiplication, while the latter is not.