I've just started to learn Noetherian rings and struggling to understand. So, I learned
A commutative ring R is Noetherian iff it satisfies the ascending chain condition on ideals iff it satisfies the maximum condition on ideals (maximum condition: every non-empty set of ideals of R contains a maximal element)
Also, I googled the non-example of Notherian rings and found that the ring of polynomials
Consider the ring of polynomials $\mathbb R[x_1,x_2,...]$ of inifintely many variables. Consider $$(x_1)\subset (x_1,x_2)\subset ...\subset (x_1,x_2,...,x_n)\subset...$$ This infinite chain of ideals does not stablize. Hence $\mathbb R[x_1,x_2,...]$ is not Noetherian. (http://mathonline.wikidot.com/noetherian-rings)
So $\mathbb R[x_1,x_2,...]$ must not satify the maximum condtion on ideals. I think $$J=\{(x_1,x_2,...,x_k):k\in\mathbb N \} $$does not satisfy the maximum condition. However, I also learned that every ideal in a commutative ring with unity always have a maximal ideal and every ideal in R is contained in a maximal ideal. But then what is the maximal ideal of $\mathbb R[x_1,x_2,...]$?