Suppose our ring $R = k[t]$. Let $M$ be the $R$-module $R[x]/(tx - t)$. Prove that $M$ isn't flat.
73662
Example of non-flat modules
67633
Generic flatness on modules
102072
Simple $M_n(D)$-module with $D$ a division ring
151803
Prove that $R$ is commutative
9806
$M$ finitely generated if submodule and quotient are finitely generated.
98818
Quotient ring of a localization of a ring
4507
Localization of a ring which is not a domain
124864
A not free $\mathbb{Z}$-module
80302
A module $M$ whose submodules and factor modules are semisimple but not semisimple itself
85032
$Q(\mathbb{Z}[t]) / \mathbb{Z}[t]$ is not injective
116883
About ideals in a ring which is a $k$ vector space
133238
Why does $\mathrm{Tor}_0^R(M,N)\cong M\otimes_R N$?
156679
$K[X,Y]/(XY-1)$ not isomorphic to $K[T]$
42871
Quotient of a local regular ring
155642
Understanding divison by monic polynomial in $R[x]$ where $R$ is an arbitrary ring
128861
If $M$ isn't Noetherian, $M$ has a submodule maximal with respect to being not finitely generated.
156457
Show that $K[x,xy,xy^2,\dots]$ is not Noetherian