How to finish this argument showing preimage of maximal ideal is maximal under surjective map.
67237
Pull back image of maximal ideal under surjective ring homomorphism is maximal
100560
$X$ compact Hausdorff space, characterize the maximal ideals of $C(X)$
53770
Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime.
116000
Prove that prime ideals of a finite ring are maximal
19230
Prime/Maximal Ideals in $\mathbb{Z}[\sqrt d]$
40678
Image of an ideal under a surjective ring homomorphism is an ideal
110022
Showing prime and maximal ideal
63447
Prove that the kernel of a homomorphism is a principal ideal. (Artin, Exercise 9.13)
12490
A map is continuous if and only if for every set, the image of closure is contained in the closure of image
16092
How to prove that the composition of two surjective functions is surjective
57198
How to show that a valuation ring has a unique maximal ideal?
45792
Rigorous proof that surjectivity implies injectivity for finite sets
131497
Maximal ideal not containing the set of powers of an element is prime
28244
Is the ideal $I = \{f\mid f (0) = 0\}$ in the ring $C [0, 1]$ of all continuous real valued functions on $[0, 1]$ a maximal ideal?
68202
Must an ideal generated by an irreducible element be a maximal ideal?
49521
In an extension of finitely generated $k$-algebras the contraction of a maximal ideal is also maximal
93641
A maximal ideal among those avoiding a multiplicative set is prime