I read here that if $R$ is an integral domain and has ACCP, then $R[x]$ also has ACCP. However, is this necessarily true for a commutative ring with identity? If it is false, then what is a counterexample?
I have tried constructing a counterexample by taking $a,b \neq 0 \in R$ such that $ab=0$ e.g. they are zero divisors, and trying to construct an ascending chain of polynomial generators that increase in degree, e.g. $P_{i+1}\mid P_i$ but $\deg (P_{i+1})>\deg (P_i),$ since the argument used in the linked post used the fact that the degree must stabilize.
However, every time I increase the degree (by "dividing" by some polynomial with terms of $a$ or $b$), it seems like the exponent of $a$ and $b$ increase, leading me to think that if we start with exponent $k,$ eventually it will reach $1$ and again, the degree will stabilize.
