Interpreting statements in Lang's Undergraduate Algebra So, I've been reading this book and I've come across two sentences that I find a little confusing.

On pg. 109:
The polynomial ring $R[t]$ is generated by the variable $t$ over $R$, and $t$ is transcendental over $R$.
Context: $R[t]$ is the polynomial ring, and, for a fixed $x \in R$,  $R[x] = \{ f(x) : f \in R[t] \}$. $x$ is transcendental if $f \mapsto f(x)$ is an isomorphism from $R[t]$ to $R[x]$.
Problem: $t$ doesn't seem to be an element of $R$.

On pg. 117:
Let $F$ be a field and $\sigma : F[t] \rightarrow F[t]$ is an automorphism of the polynomial ring such that $\sigma$ restricts to the identity on $F$.
Problem: This seems to act as if $F \subset F[t]$?

Thanks the help.
 A: From the comments above.

For the first question:
There is a problem with the definition you have of a transcendental element. You say that

[F]or a fixed $x\in R$, $R[x]=\{f(x):f\in R[t]\}$ [and] $x$ is transcendental if $f\mapsto f(x)$ is an isomorphism from $R[t]$ to $R[x]$.

I take it that $R[x]=\{f(x):f\in R[t]\}$ is meant to be a definition of $R[x]$, in which case it is quite restrictive. What should be said is that if $R$ and $S$ are rings with $R \subset S$, then for any fixed $x \in S$ we define $R[x] = \{ f(x) : f \in R[t] \}$. Now, $x \in S$ is said to be transcendental if $f \mapsto f(x)$ is an isomorphism from $R[t]$ to $R[x]$.
The earlier definition defined transcendence (over $R$) only on elements of $R$, which is inappropriate because no element of $R$ is transcendental over $R$.
For the second question:
There is a natural inclusion of $F$ inside $F[t]$, given by identifying $a \in F$ with the constant polynomial $at^0$. In this way we can view $F$ as a subset of $F[t]$.
