HOw I can start this ?Let M be an R-module and e : M → M a morphism of R-modules such that e 2 = e. Show that M = ker(e) ⊕ Im(e).
141149
Proof some module is direct sum by its submodules.
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?
30327
$\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$
151803
Prove that $R$ is commutative
80211
Annihilator of quotient module M/IM
140340
Left and right R-modules
137029
Prove that $\operatorname{Hom}_{\Bbb{Z}}(\Bbb{Q},\Bbb{Z}) = 0$ and show that $\Bbb{Q}$ is not a projective $\Bbb{Z}$-module.
55148
Why is this isomorphism $M \otimes_K L \stackrel{\simeq}{\longrightarrow} M^{[L:K]}$ an isomorphism of $M$ - algebras?
114916
$\mathbb Z_{mn}$ isomorphic to $\mathbb Z_m\times\mathbb Z_n$ whenever $m$ and $n$ are coprime
144122
A problem about an $R$-module that is both injective and projective.
4295
Let $N$, $K$ be sub-modules of $M$ with $I=\mathrm{Ann}(N)$, $J=\mathrm{Ann}(K)$. Show $I+J$ is a proper subset of $\mathrm{Ann}(N \cap K)$.
70872
Isomorphism of quotient modules implies isomorphism of submodules?
137011
showing $\psi: R\to \mathbb C$ is ring isomorphism.
40705
If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$ an ideal, then $M$ has a structure of $R/I$-module
72295
A difficulty in understanding the proof of distributivity of tensor products over direct sums for modules.
103349
A question about a proof of Noetherian modules and exact sequences
123130
Showing an isomorphism between exterior products of a free module