Inverse image of maximal ideal under $k$-algebra homomorphism is a maximal ideal.
49521
In an extension of finitely generated $k$-algebras the contraction of a maximal ideal is also maximal
109691
When are kernels of homomorphisms on $\mathbb{Z}[x]$ principal ideals, and what does this have to do with Gauss's lemma?
46040
Preimage of maximal ideal is maximal (why is this answer wrong?)
44065
If $B$ is finitely generated as a $k$-algebra, and $\phi:A\to B$ is a $k$-algebra map, is $\phi^{-1}(M)$ maximal for any maximal $M\subset B$?
138130
If $I$ is a maximal ideal of $R$, why is $R/I$ a field?
131943
How to directly prove that $M$ is maximal ideal of $A$ iff $A/M$ is a field?
99567
Proving that an inverse ring homomorphism of an ideal is an ideal?
64025
Ring homomorphisms from $\Bbb Q$ into a ring
121244
Homomorphisms and automorphisms on polynomial rings
464
Prove that the ideal $(X_1-a_1,...,X_n-a_n)$ is maximal in $K[X_1,\dots,X_n]$
74537
Maximal Ideals in $K[y]$
30327
$\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$
49948
If every ideal is principal show the same is true for the image of a homomorphism.
63447
Prove that the kernel of a homomorphism is a principal ideal. (Artin, Exercise 9.13)
48818
Homomorphism of local rings
118306
$I:=\{f(x)\in R\mid f(1)=0\}$ is a maximal ideal?
136654
$(x_1-a_1, x_2-a_2)$ is a maximal ideal of $K[x_1,x_2]$