Let R be a commutative ring. Let A 0 be an ideal in R[x] such that
the lowest degree of a nonzero polynomial in A is n 1 and such that A
contains a monic polynomial of degree n. Prove that A is a principal ideal
in R[x].
let be a monic polynomial of degree obviously now using induction over degrees of elements of we show that every elment of is divisible by which will
prove that so let then is a polynomial of degree at most and thus by the minimality of we have
thus this proves the base of the induction. now let and suppose every polynomial in of degree at most is in
let then and thus by induction hypothesis for some then