I'm given the following definitions of an $A-$algebra over a commutative ring with 1 and ring extension: an $A-$algebra $B$ is a ring $B$ together with a given ring homomorphism $\phi: A\rightarrow B$. If $A\subset B$, then $B$ is called an extension ring of $A$. A few questions arise:
- How does one know what the map $\phi$ is? To be specific, the author of the book I'm using states that $k[x^2]\subset k[x]$ is an (integral) extension. By the above definition it means in particular that $k[x]$ is a $k[x^2]-$algebra, so it is assumed that there is a homomorphism of rings $\phi: k[x^2]\rightarrow k[x]$. How am I supposed to understand the formula by which this $\phi$ is defined?
- I read somewhere that often $\phi$ is the inclusion map. What are simple examples when it is not?