The subring of $k[x, y]$ generated by $\{x y^{i}: i\geq 0\}$ is not finitely generated over $k$
70712
How to prove that $k[x, xy, xy^2, \dotsc]$ is not noetherian?
155604
Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable.
55026
$\mathbb{Q}[X,Y]/(Y^2-X^3)$ is not a UFD
124864
A not free $\mathbb{Z}$-module
22529
The support of a non finitely generated module
85032
$Q(\mathbb{Z}[t]) / \mathbb{Z}[t]$ is not injective
13606
Prove that $K$ is isomorphic to a subfield of the ring of $n\times n$ matrices over $F$.
87534
Finitely generated group which is not finitely presented
156443
if $x$ and $y$ commute with $[x y]$, then $[x y]$ is nilpotent
85987
Prove that $k[x,y,z,w]/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain
31142
Why does $K \leadsto K(X)$ preserve the degree of field extensions?
143429
Kernel of ring homomorphism $k[X,Y] \rightarrow k[t^2, t^3]$
75970
$\mathbb Z[1/2]$ is not finitely generated?
9794
Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime
151786
Proving that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}$
93474
A Finite Boolean Ring is Generated by Finitely Many Copies of $\mathbb{Z} / 2 \mathbb{Z}$
31451
Proving a subring of $\mathbb{Q}$ containing $\mathbb{Z}$ is a PID