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Questions tagged [direct-sum]

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

Direct sums are defined for a number of different sorts of mathematical objects, including subspaces, matrices, modules, and groups.

Definition: Let $~U,~ W~$ be subspaces of $~V~$ . Then $~V~$ is said to be the direct sum of $~U~$ and $~W~$, and we write $~V = U ⊕ W~$, if $~V = U + W~$ and $~U ∩ W = \{0\}~$.

  • The significant property of the direct sum is that it is the coproduct in the category of modules (i.e., a module direct sum).
  • Direct products and direct sums differ for infinite indices. An element of the direct sum is zero for all but a finite number of entries, while an element of the direct product can have all nonzero entries.
  • The direct sum of Abelian groups is the same as the group direct product, but that the term direct sum is not used for groups which are non-Abelian.

References:

https://en.wikipedia.org/wiki/Direct_sum

http://mathworld.wolfram.com/DirectSum.html

1119 questions
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Direct Sum vs. Direct Product vs. Tensor Product

There are a lot of questions like this all over the site, but I cannot find one that resolved my confusion- what are the formal definitions of direct sums, direct products, and tensor products (in the most general sense), and how are they different?
user247773
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3 answers

Difference between sum and direct sum

What is the difference between sum of two vectors and direct sum of two vector subspaces? My textbook is confusing about it. Any help would be appreciated.
Marion Crane
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About direct sum of abelian groups and quotient

I'm trying to understand properly the relations between quotient and direct sum. The first thing I wanted to know, and couldn't find online, is whether my guess is true or not: Assume $G_\alpha$ are abelian groups, and $H_\alpha \leq G_\alpha$ a…
TopologyGetsMeTired
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3 answers

The definition of direct sums and subspaces?

From what I have read the definition of the direct sum of the vector spaces $V_1$, $V_2$ is the set $V_1\times V_2$ with the operations of addition and scalar multiplication defined as follows (Knapp,…
Quantum spaghettification
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Visual intuition for direct sum vs. tensor product of vector spaces

I completely understand the formal mathematical distinction between the direct sum and the tensor product of two vector spaces. I also understand that the direct sum has a nice visual interpretation (especially the direction sum of two 1D vector…
tparker
  • 6,756
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Localization commutes with arbitrary direct sums

Let $M_i$ be a arbitrary colection of $A$-modules and $S$ a multiplicative subset of $A$. I want to show that $$S^{-1}\left(\bigoplus_i M_i\right)\cong \bigoplus_i S^{-1}M_i$$ as $A$-modules and as $S^{-1}A$-modules. I know how to explicitly write…
Gabriel
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Direct sums and direct products

This question has been in my head for a while. And today it appears again when I am reading Arveson's book on $C^*$-algebras. He says Countable direct products of Polish spaces are Polish. Countable direct sums of Polish spaces are…
Hui Yu
  • 15,469
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Direct sum and direct product of infinitely many abelian groups are not isomorphic

Let $I$ be an infinite set, and for each $i$ let $A_i$ be an abelian group with order $o(A_i) \ge 2$. Prove that the direct product $\prod A_i$ and the direct sum (coproduct) $\bigoplus A_i$ are not isomorphic. Here the product and coproduct are…
Caleb Stanford
  • 47,093
13
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1 answer

Module which is not direct sum of indecomposable submodules

I would like to find an example of a ring $R$ and a $R$-module $M$, which can't be written as a direct sum of indecomposable submodules, i.e. $$ M \not \cong \bigoplus\limits_{i \in I} M_i$$ for all set $\{M_i \;\vert\; i \in I \}$ of…
Watson
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Does $A\oplus \mathbb{Z}\cong B\oplus \mathbb{Z}$ imply $A\cong B$?

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…
Cookie Monster
  • 943
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Dual of $\ell^p$ Direct sum

I am asked to show that the $\ell^p$-direct sum of a sequence of Banach Spaces $X_n$ is isometrically isomorphic to the $\ell^q$ direct sum of $X_n^*$ where $X_n^*$ is the dual of $X_n$ for each $n$ respectively. (Here $p$, $q$ are conjugate indices…
Jack
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Prove vector space is the direct sum of subspace and its orthogonal complement

$V$ is finite-dimensional over $\Bbb{C}$ and the form $\langle \cdot , \cdot \rangle$ is Hermitian. $U$ is a subspace of $V$. Show that $V = U \oplus U^\perp$ I've been able to show that $U \cap U^\perp = \{0\}$. I don't know how to approach the…
mathsishard
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Why is cohomology the direct product of the $H^n$?

During a talk I mentioned in passing Borel's result that for $G$ a connected Lie group, $H^*(BG;\mathbb Q)$ is a polynomial ring. An audience member corrected me in very short order that no, it's in fact a power series ring. I responded that while…
jdc
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Are there any non-obvious colimits of finite abelian groups?

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…
Martin Brandenburg
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distributivity of tensor product and direct sum for Hilbert spaces

Before I ask my actual question about direct sums and tensor products of Hilbert spaces, let's first talk about direct sums and tensor products of vector spaces. We might define direct sums of vector spaces by the corresponding universal mapping…
Corsin Pfister
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