$(X^m-Y^n)$ is a prime ideal in $A[X,Y]$ iff $m, n$ are coprime
2043
$X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$
116000
Prove that prime ideals of a finite ring are maximal
53770
Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime.
145594
Splitting of prime ideals in algebraic extensions
137858
Showing that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal, and also finding its cardinality?
156671
Necessary and sufficient condition for $x^n - y^m$ to be irreducible in $\Bbb C[x,y]$
97740
Homogeneous ideals are contained in homogeneous prime ideals
149747
Prove that if I is a prime ideal of R then I[x] is a prime ideal of R[x]
142863
Show that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$
63171
Translate of an ideal by $a$ coincides with the ideal iff  $a \in I$
135682
Are distinct prime ideals in a ring always coprime? If not, then when are they?
56800
Intersection of prime ideals
117786
Show that some monomial ideal is primary
56278
$I$ and $J$ are coprime ideals iff $x \to (x + I, x + J)$ is surjective.
120149
Is the complement of a prime ideal closed under both addition and multiplication?
69710
Is it true that an ideal is primary iff its radical is prime?
80232
Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is an integer $n > 1\mid a^n =a$. Every prime ideal is maximal?