Let $F$ be a field. Why is $\frac{F[x]}{(x-1)}\oplus \frac{F[x]}{(x-1)}$ not isomorphic to $\frac{F[x]}{(x-1)^2}$ as $F[x]$-modules?
102282
Let $R$ be a PID, and let $\pi\in R$ be prime. Is it $R/\langle\pi\rangle \times R/\langle \pi\rangle$ a cyclic $R$-module?
59882
is it possible for two fields with the same characteristic to not be isomorphic?
98587
$\mathbb Z [X] / (X)$ isomorphic to $\mathbb Z[X] / (X+1)$ isomorphic to $\mathbb Z [X] / (X+2015)$
81197
Why is $SO(3)\times SO(3)$ isomorphic to $SO(4)$?
127000
When does $V = \ker(f) \oplus \operatorname{im}(f)$?
106091
Showing $\mathcal{P}(X)$ is isomorphic to $2^X$
70872
Isomorphism of quotient modules implies isomorphism of submodules?
69384
Are the following modules free and/or isomorphic?
41810
Should all polynomial functions from F to F (F is a field) have coefficients in F?
85096
Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.
42810
Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?
42552
Localisation is isomorphic to a quotient of polynomial ring
89271
If $F$ is a field, then $F[x]$ is a principal ideal domain proof question
29359
Localisation isomorphic to a quotient of polynomial ring
119305
Is the splitting field equal to the quotient $k[x]/(f(x))$ for finite fields?
109610
Is $\mathbb{R}[X]/(P)$ isomorphic to $\mathbb C$ for every irreducible polynomial $P$ of degree $2?$
149849
Let $F$ be a field and $f: \mathbb{Z} \to F$ be a ring epimorphism.