This is a remark in Exercise 1.3.D in Vakil's Foundations of Algebraic Geometry:
Recall: the data of "an $A$-algebra $B$" and “a ring map $A \to B$ are the same.)
I tried to make sense of this and tried the following:
- If we have a ring homomorphism $\varphi: A \to B$, we obtain an $A$-algebra $B_\varphi = (B,+,\cdot,\times)$ whose scalar multiplication $\times$ is defined by $$ a \times b := \varphi(a) \cdot b \ $$ (here, $+$ resp. $\cdot$ stand for the ring addition resp. multiplication in $B$).
- If we have an $A$-algebra $B$, we obtain a ring homomorphism $\varphi_B: A\to B$ by setting $$\varphi_B(a) := a \times 1_B$$ (here, $1_B$ stands for the multiplicative identity element in $B$).
These two maps are inverses of each other if my calculations are correct. Could you please confirm if my approach is correct?
Thank you in advance!