I am confused with relatively prime, comaximal ideals (sum of the two ideals is the full ring) in polynomial ring in $n$ variables. Are the following statements true?
1) Let $I=(f(x,y,z))$ and $J=(g(x,y,z))$ be two principal ideals (in a polynomial ring $K[X,Y,Z]$) are relatively prime iff there does not exist any irreducible polynomial dividing both $f$ and $g$.
2) If $I$ and $J$ are relatively prime then $I+J=K[X,Y,Z]$, i.e., they are comaximal.
Are the above two statements true?
I encountered this problem when I started to compute radical of ideal $I=(XY+YZ+ZX,XYZ)$.
Please explain to me concept of relatively prime and comaximal ideals in a polynomial ring.
Best Answer
Elements $a,b$ are coprime $\,\overset{\rm def}\iff\, c\mid a,b\,\Rightarrow\, c\mid 1,\,$ i.e. they have only unit common divisors.
Ideals $A,B$ are comaximal $\,\overset{\rm def}\iff\, A + B = 1$.
In a PID like $\Bbb Z$ we have $(a,b) = (c) \!\iff\! c = \gcd(a,b),\,$ so $\,(a,b)=1\!\iff\! \gcd(a,b) = 1,\,$ i.e. $(a),(b)$ are comaximal $\!\iff\! a,b\,$ are coprime. However, this in not generally true in non-PIDs, $\,$ e.g. $\,(X),\, (Y)$ are coprime, not comaximal, by $(X)+(Y)= 1 \overset{\exists f,g}\Rightarrow\ f X + g Y = 1\overset{X = 0 = Y}\Rightarrow 0 = 1$.
More generally, for a UFD (Unique Factorization Domains) we have
Theorem $\rm\ \ \ TFAE\ $ for a $\rm UFD\ D$
$(1)\ \ $ prime ideals are maximal if nonzero, $ $ i.e. $\rm\ dim\,\ D \le 1$
$(2)\ \ $ prime ideals are principal
$(3)\ \ $ maximal ideals are principal
$(4)\ \ \rm\ gcd(a,b) = 1\, \Rightarrow\, (a,b) = 1, $ i.e. $ $ coprime $\Rightarrow$ comaximal
$(5)\ \ $ $\rm D$ is Bezout
$(6)\ \ $ $\rm D$ is a $\rm PID$
Proof $\ $ (sketch of $1 \Rightarrow 2 \Rightarrow 3 \Rightarrow 4 \Rightarrow 5 \Rightarrow 6 \Rightarrow 1)\ $ where $\rm\,p_i,\,P\,$ denote primes $\neq 0$
$(1\Rightarrow 2)$ $\rm\ \ p_1^{e_1}\cdots p_n^{e_n}\in P\,\Rightarrow\,$ some $\rm\,p_j\in P\,$ so $\rm\,P\supseteq (p_j)\, \Rightarrow\, P = (p_j)\:$ by dim $\le1$
$(2\Rightarrow 3)$ $ \ $ max ideals are prime, so principal by $(2)$
$(3\Rightarrow 4)$ $\ \rm \gcd(a,b)=1\,\Rightarrow\,(a,b) \subsetneq (p) $ for all max $\rm\,(p),\,$ so $\rm\ (a,b) = 1$
$(4\Rightarrow 5)$ $\ \ \rm c = \gcd(a,b)\, \Rightarrow\, (a,b) = c\ (a/c,b/c) = (c)$
$(5\Rightarrow 6)$ $\ $ Ideals $\neq 0$ in Bezout UFDs are generated by an elt with least #prime factors
$(6\Rightarrow 1)$ $\ \ \rm (d) \supsetneq (p)$ properly $\rm\Rightarrow\,d\mid p\,$ properly $\rm\,\Rightarrow\,d\,$ unit $\,\rm\Rightarrow\,(d)=(1),\,$ so $\rm\,(p)\,$ is max