let be an A-algebra and let then there are 3 equivalent conditions:

i) is integral over A

ii)the subring generated by is finite over A (as an A-module?)

ii) an A-algebra such that and C is finite over A.

Proof:

since y is integral then we know that there exists a monic polynomial s.t (*) but then the proof says is generated by , as an A-module.

To show this let

if then substracting times the relation (*) kills the leading term......I cannot get to this.... please help

how to show ?

can I just set ?? then and (every ring is a subring of itself right?)