Let $R$ be a ring and $A,B$ two simple $R$-modules. I would like to prove the following: If $A$ and $B$ are not isomorphic, then the only submodules of $A \times B$ are $\{0\} \times \{0\},A \times \{0\}, \{0\} \times B$ and $A \times B$. The result is wrong in general (see this question). Let […]

In the books 1 and 2, in Somme directe d’une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if $$\forall e \in E, f \in F (e+f=0_V \to e=f=0_V)$$ 2) let $E,F$ two vector subspaces of $V$, then […]

Let us consider the ring $ R:=\begin{bmatrix}\Bbb Z & 0\\ \Bbb Q & \Bbb Q\end{bmatrix} $ and its two-sided ideal $ D:=\begin{bmatrix}0 & 0\\ \Bbb Q & \Bbb Q\end{bmatrix} $. Let then consider the free right $R$-module $F_R:=\bigoplus_{\lambda\in\Lambda}x_{\lambda}R$. I must show that $$ \bigcap_{n\ge1}nF_R=\bigoplus_{\lambda\in\Lambda}x_{\lambda}D=F_RD\;\;. $$ I proved the first equality using the fact that $\bigcap_{n\ge1}nR=D$. […]

When first learning about tensor products and direct sums of vector spaces, the analogy of multiplication and addition of vector spaces is sometimes used to help with the intuition. If we look at the bases of the considered vector spaces, this analogy seems most intuitive: the basis of the direct product of $U$ and $V$ […]

If $A$ and $B$ are abelian groups, do we have that $A\oplus \mathbb{Z}\cong B\oplus \mathbb{Z}$ implies $A\cong B$? Motivation: I was just thinking about different ways of deducing equality from expressions by quotienting, then realized I didn’t know the answer in this case.

Does the forgetful functor $U : \mathsf{FinAb} \to \mathsf{Ab}$ from finite abelian groups to abelian groups preserve colimits? Morally this should be true, but it is not so easy (for me) to come up with a nice proof. If $(A_i)$ is a diagram of finite abelian groups, assuming that their colimit $\mathrm{colim}_i A_i$ in $\mathsf{FinAb}$ […]

Given the language $\{+,0\}$, I would like to prove that $\mathbb{Z} \oplus \mathbb{Z}$ is not isomorphic to $\mathbb{Z}$, but I am not sure where to start?

Is it true that if $ U \oplus W_1 = U \oplus W_2 $, then $ W_1 = W_2 $? I think that if $ U \oplus W_1 = U \oplus W_2 $, then u+w1=u+w2, so W1=W2. But did I make any mistakes?

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