Is This an Interesting Principal Ideal?

Question

Let $I$ be a principal ideal in a commutative ring $R$, $x$ the generator of $I$, and $x^n\ne \sum_{i=0}^{n-1}a_ix^i,\, \forall a_i\in R\setminus I\bigcup\{0\},\, \forall n\in\mathbb N$. In other words, for this particular principal ideal, any integer power of the generator is assumed to not be expandable in its polynomial of lower orders with coefficients in the ring but not in the principal ideal. Is there a name and special interest in such an object? What are some of the interesting sufficient conditions that can achieve the aforementioned non-expandability condition?

Answer 1

Let $R$ be an integral domain and let $I$ be a principal ideal with $x$ s.t. $I = xR$. If $I$ does not have the property proposed in question, there exist the smallest natural number $n$ such that $a_i \in R$ and $x^n = \sum_{i=0}^{n-1}a_ix^i \iff x^n - \sum_{i=1}^{n-1}a_ix^i = x(x^{n-1} - \sum_{i=0}^{n-2}a_{i+1}x^i) = a_0$. Now $(...)\ne 0$ by $n$ being the smallest such natural number, together with $x\ne0$, we have $a_0\ne0$ as $R$ is an integral domain. But then the LHS $\in I$ and the RHS $\notin I$, a contradiction. The conclusion is that in an integral domain every principal ideal has the proposed property.