Show that the ring of polynomials with coefficients in a field, and in infinitely many variables, is not Noetherian Show that the ring of polynomials with coefficients in a
field, and in infinitely many variables, is not Noetherian, that is, $R = k [x_i: i\geq1]$ is not Noetherian.
I know that I need to exhibit an ideal of the ring that is not finitely generated, what could this ideal be? Could it be $(x_1,x_2,...,)$? 
Or could I give the following chain of ideals that do not have a maximal element $(x_1)\subset(x_1,x_2)\subset(x_1,x_2,x_3)\subset...$?How can all ideals that are not finitely generated be classified? What to do in the case where the number of variables is non-countable?

A ring is Noetherian if and only if it satisfies the ascending chain condition, i.e. every increasing chain of ideals terminates. Now you have a chain 
$$(x_1)\subsetneq(x_1,x_2)\subsetneq(x_1,x_2,x_3)\subsetneq\cdots$$
that never terminates, so $k[x_i: i\ge 1]$ is not Noetherian.
