If $A$ is a noetherian ring and $a$ is an ideal of $A$, then $A$ and $A/a$ are not isomorphic rings.
70320
Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is $I=(0)$?
104562
Prove that $\mathbb{Z}$ is not isomorphic to additive group of any vector space over any field.
153291
Why a ring homomorphism from a local ring will not map a non-unit to a unit？
127030
Simple example of non-arithmetic ring (non-distributive ideal lattice)
156679
$K[X,Y]/(XY-1)$ not isomorphic to $K[T]$
108955
Showing $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$
97608
Showing a Ring of endomorphisms is isomorphic to a Ring
151786
Proving that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}$
137285
Prove $\mathbb{Z}_2 \times \mathbb{Z}_2$ is not isomorphic to $\mathbb{Z}_4$
88645
Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic.
140865
Rings' second isomorphism theorem
140553
An ideal which is not finitely generated
62917
Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura
103349
A question about a proof of Noetherian modules and exact sequences
122012
The ideal $(x,y)$ is not a free $K[x,y]$-module
8587
What is an example of two k-algebras that are isomorphic as rings, but not as k-algebras?
8694
$A_4$ is not isomorphic to $D_6$