# Questions tagged [direct-sum]

For questions about taking the direct sum of groups and other algebraic structures.

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### Does the direct sum have a universal property in the category of groups?

In other categories, like modules, the direct sum is the coproduct, giving it a neat category theory description. The direct product of groups, which is very similar to it, is the product in the ...
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### Decomposition of TTM into HM and VM

Why is it that if I have a smooth manifold and a connection map $K$, defined below, is it the case that it induces a decomposition of the tangent space to the tangent space to the manifold, given as ...
1 vote
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### Are there internal direct products of vector spaces?

When I learned linear algebra I was a bit confused about the notation $A \oplus B$, the direct sum of two vector spaces $A, B$ over some field $K$. As a set, $A \oplus B$ is nothing else than the ...
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### $N_i$'s independent $\iff$ $\oplus_i N_i\cong\sum_i N_i$?

Notation and conventions: I will use $\oplus$ for the "external direct sums" only. All the modules will be over a commutative ring with unity, $A$. Problem: In the proof of one of the ...
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1 vote
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### Theorem 5, Section 7.2 of Hoffman’s Linear Algebra

Definition: $A\in M_{n\times n}(F)$ is in rational form if $$A=\begin{bmatrix} A_1& & \\ & \ddots & \\ & & A_r\\ \end{bmatrix}$$ where $A_i$ is companion matrix of non scalar ...
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1 vote
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### Direct Sum of Topological $R$-modules

Please help me. I am proving the following property: Let $R$ be a topological ring and $\{A_{\lambda}\}_{\lambda\in\Lambda}$ be a family of topological $R$-modules. Let $B_{\lambda}$ be an ...
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### Is a group complement of a subspace of a vector space necessarily a subspace?

Let $W$ be a vector space. Let $U$ be a subspace and $V$ be a subgroup of $W$, and assume that $W$ is the direct sum (as groups!) of $U$ and $V$. Does it follow that $V$ is a subspace, such that in ...
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### Endomorphisms between direct product/sum of algebraic objects?

My motivation is this Wikipedia article. Suppose $R$ be a ring with unity and $M,N$ be $R$-modules. Take their direct sum/product $P=M \oplus N$. So $P$ is also a $R$-module. Consider the respective ...
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### The functor $\mathrm{Hom}(A,-)$ cannot commute with arbitrary direct sums for infinitely generated projective module $A$

It is easy to see the functor $\mathrm{Hom}(A,-)$ commutes with every arbitrary direct sum (i.e. $\mathrm{Hom} (A,\oplus_{i\in I} N_i)=\oplus_{i\in I}\mathrm{Hom}(A,N_i)$) for finitely generated ...
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### Theorem 11, Section 6.7 of Hoffman’s Linear Algebra

Let $T$ be a linear operator on a finite-dimensional space $V$. If $T$ is diagonalizable and if $c_1,…, c_k$ are the distinct characteristic values of $T$, then there exist linear operators $E_1,…,E_k$...
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