What does the following ideal look like

Question

Let $R$ be a commutative ring with 1. What does the principle ideal generated by $x$ in the polynomial ring $R[x]$ look like ?

So what I'm asking is which of the following is right definition.

$(x) := \{rx : r\in R[x]\}$

or

$(x) := \{rx : r\in R\}$

Answer 1

The second one is not even closed under multiplication (consider $x^2$). The first one is correct.

For any commutative ring with identity, $(x)=xR$. In this case, your ring is $S=R[x]$, so it ought to be $xS$.