$F$ a field, $K_{i}$ a subfield of $F$, $P = \cap K_{i}$ (P is a prime subfield) and $R$ a prime ring of $F$
"... since $1 \in R$, we have $k.1 \in R$ for all $k \in \mathbb{Z}$. Therefore, $R \subset P$..."
This is in my professor's class notes. I didn't understand why $R \subset P$. Can anybody help me?
It's hard to tell without knowing exactly what your definitions are.
The prime ring, being the smallest subring of $F$, is obviously contained inside of $P$, the smallest subfield of $F$.
Or another way: the prime ring is generated by the identity $1$. All the subfields of $F$ share the same identity, so they all contain $1$, and that means $P$ contains $1$. Since $P$ is a ring and contains $1$, it must contain the prime subring.