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Universal property of a direct sum of modules

Statement

Let $(M_{i})_{i \in I}$ be left $R$ -modules and $N$ a left $R$ -module. For every family of homomorphisms $f_{i} : M_{i} \to N$ , there is a unique homomorphism $f : ⨁_{i \in I} M_{i} ⟶ N$ such that $f \circ ȷ_{i} = f_{i}$ for every $i$ . It is given by $f ((m_{i})) = \sum_{i \in supp (m)} f_{i} (m_{i}) .$ For $I = \emptyset$ , this is the unique map $0 \to N$ .

Facts & Assumptions

Given: A family $(M_{i})_{i \in I}$ of left $R$ -modules, a left $R$ -module $N$ , and homomorphisms $f_{i} : M_{i} \to N$ .

[F1]

Elements of $⨁_{i} M_{i}$ have finite support, and $ȷ_{i}$ is the coordinate inclusion (The direct sum of an indexed family of modules).

[F2]

A module homomorphism preserves addition and scalar multiplication (Module homomorphism and isomorphism, kernel, image and cokernel).

Proof

technique · constructive

1.1F1construct

Define $f ((m_{i})) : = \sum_{i \in supp (m)} f_{i} (m_{i})$ ; the sum is finite by [F1], and padding it by zero terms shows it is independent of the chosen finite set containing the support.

2.1step 1.1F1F2

Addition and scalar multiplication may be checked on the finite union of the relevant supports, so [F2] gives $f (m + m^{'}) = f (m) + f (m^{'})$ and $f (r m) = r f (m)$ .

2.2step 1.1F1

For $x \in M_{i}$ , step 1.1 gives $f (ȷ_{i} (x)) = f_{i} (x)$ , so $f \circ ȷ_{i} = f_{i}$ .

2.3step 1.1F1F2

Every $m = (m_{i})$ equals the finite sum $\sum_{i \in supp (m)} ȷ_{i} (m_{i})$ . Hence any homomorphism $g$ satisfying $g \circ ȷ_{i} = f_{i}$ has $g (m) = \sum_{i} f_{i} (m_{i}) = f (m)$ and therefore equals $f$ .

3.1step 1.1step 2.1step 2.2step 2.3F1discharge-construct∎

If $I = \emptyset$ , the direct sum is $0$ by [F1], the formula is the empty sum, and the construction and uniqueness still apply. Thus the universal property holds for every index set.

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)