Why do the polynomials $x^5$ and $x^6$ in the ring of polynomials without a linear term have no greatest common divisor? Why isn't it $x^3$?
Hint $\,\ x^2\mid x^5,x^6\ $ but $\, x^2\nmid x^3\ $ since $\ x^3/x^2 = x\not\in R$
Beause in $k[x^2, x^3]$ both $x^2$ and $x^3$ are divisors of $x^5$ and $x^6$. If $x^3$ were a gcd, it must me a multiple of $x^2$, which is not true. The set of common divisors does not have a largest element with respect to the $|$-order.