Notation: If $K$ field, $t$ transcendent over $K$, then what is $K[t]$?

Question

I'm a little bit confused by the notation "$K[t]$", what does it denote? I know that $K[X]$ is the polynomial ring over $K$, so the elements $P$ of $K[X]$ look like $P(X) = \sum_{n=0}^\infty k_n X^n$ with $k_n \in K$. Is $K[t]$ then just the image of the homomorphism $P \mapsto P(t)$ with $P \in K[X]$? Wouldn't that just be same as $K(t)$, which is the smallest field that contains both $K$ and $t$? What's the difference?

Answer 1

$K(t)$ is the smallest field that contains $K$ and $t$, but $K[t]$ is the smallest ring that contains $K$ and $t$. For example, $t^{-1}$ is in $K(t)$ but not in $K[t]$.