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A construction extensively used in theories of mathematical structures which form categories similar to an [[Abelian category|Abelian category]]. In the non-Abelian case the direct sum is usually called the discrete direct product. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d0327401.png" /> be a class of single-sorted algebraic systems which contain the one-element (zero) subsystems. The direct sum, or (discrete) direct product, of systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d0327402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d0327403.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d0327404.png" /> is the subsystem of the [[Direct product|direct product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d0327405.png" /> consisting of those functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d0327406.png" /> for which all values, except for a finite number, belong to the corresponding zero subsystem. A direct sum is denoted by one of the following symbols:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d0327407.png" /></td> </tr></table>
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A construction extensively used in theories of mathematical structures which form categories similar to an [[Abelian category|Abelian category]]. In the non-Abelian case the direct sum is usually called the discrete direct product. Let  $  \mathfrak A $
 +
be a class of single-sorted algebraic systems which contain the one-element (zero) subsystems. The direct sum, or (discrete) direct product, of systems  $  X _ {i} $,
 +
$  i \in I $,
 +
in  $  \mathfrak A $
 +
is the subsystem of the [[Direct product|direct product]]  $  X = \prod _ {i \in I }  X _ {i} $
 +
consisting of those functions  $  f : I \rightarrow X $
 +
for which all values, except for a finite number, belong to the corresponding zero subsystem. A direct sum is denoted by one of the following symbols:
 +
 
 +
$$
 +
\prod _ {i \in I } {}  ^  \otimes  X _ {i} ,\ \
 +
\prod _ {i \in I } {}  ^  \oplus  X _ {i} ,\ \
 +
\sum _ {i \in I } X _ {i} ,\ \
 +
\oplus _ {i \in I } X _ {i}  $$
  
 
For a finite number of terms one also uses the notation:
 
For a finite number of terms one also uses the notation:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d0327408.png" /></td> </tr></table>
+
$$
 +
X _ {1} \dot{+} \dots \dot{+} X _ {n} .
 +
$$
  
 
The coincidence of the direct sum and the direct product in the case of a finite number of terms follows immediately from the definitions.
 
The coincidence of the direct sum and the direct product in the case of a finite number of terms follows immediately from the definitions.
  
For each term of a direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d0327409.png" /> there exists a canonical imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274010.png" /> that assigns to an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274011.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274013.png" /> takes the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274014.png" /> at the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274015.png" /> and vanishes elsewhere. Hence it can be said that a direct sum contains its terms. In the case of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274016.png" />-group (in particular, for groups, Abelian groups, vector spaces, and rings) one can give an  "intrinsic"  characterization of a direct sum. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274017.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274018.png" /> is the direct sum of a family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274019.png" />-subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274021.png" />, if: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274022.png" /> is generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274024.png" />; b) each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274025.png" /> is an ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274026.png" />; and c) the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274027.png" /> with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274028.png" />-subgroup generated by the remaining ideals is the trivial subgroup, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274029.png" />. Cf. also [[Multi-operator group|Multi-operator group]].
+
For each term of a direct sum $  X = \prod _ {i \in I }  X _ {i} $
 +
there exists a canonical imbedding $  q _ {i} : X _ {i} \rightarrow X $
 +
that assigns to an element $  x \in X _ {i} $
 +
the function $  q _ {i} ( x) : I \rightarrow X $,  
 +
where $  q _ {i} ( x) $
 +
takes the value $  x $
 +
at the argument $  i $
 +
and vanishes elsewhere. Hence it can be said that a direct sum contains its terms. In the case of an $  \Omega $-
 +
group (in particular, for groups, Abelian groups, vector spaces, and rings) one can give an  "intrinsic"  characterization of a direct sum. An $  \Omega $-
 +
group $  G $
 +
is the direct sum of a family of $  \Omega $-
 +
subgroups $  G _ {i} $,  
 +
$  i \in I $,  
 +
if: a) $  G $
 +
is generated by the $  G _ {i} $,  
 +
$  i \in I $;  
 +
b) each $  G _ {i} $
 +
is an ideal in $  G $;  
 +
and c) the intersection of $  G _ {i} $
 +
with the $  \Omega $-
 +
subgroup generated by the remaining ideals is the trivial subgroup, for each $  i $.  
 +
Cf. also [[Multi-operator group|Multi-operator group]].
  
 
Every vector space is a direct sum of one-dimensional subspaces. Every free Abelian group is a direct sum of infinite cyclic groups. Every finite Abelian group is a direct sum of cyclic groups of prime-power order. Every semi-simple associative ring with a unit element and satisfying the minimum condition for ideals is the direct sum of a finite number of complete rings of linear transformations of appropriate finite-dimensional vector spaces.
 
Every vector space is a direct sum of one-dimensional subspaces. Every free Abelian group is a direct sum of infinite cyclic groups. Every finite Abelian group is a direct sum of cyclic groups of prime-power order. Every semi-simple associative ring with a unit element and satisfying the minimum condition for ideals is the direct sum of a finite number of complete rings of linear transformations of appropriate finite-dimensional vector spaces.
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In category theory, the concept dual to that of a product, i.e. that of a [[Coproduct|coproduct]], is sometimes called the direct sum.
 
In category theory, the concept dual to that of a product, i.e. that of a [[Coproduct|coproduct]], is sometimes called the direct sum.
 
 
  
 
====Comments====
 
====Comments====
 
As already noted the direct sum is also called the discrete direct product (cf. [[Direct product|Direct product]]).
 
As already noted the direct sum is also called the discrete direct product (cf. [[Direct product|Direct product]]).
  
In category theory the direct sum or coproduct is defined by a universal property: Given objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274031.png" />, in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274032.png" />. The direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274033.png" /> is an object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274034.png" /> together with morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274035.png" /> such that for each object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274036.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274037.png" /> and family of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274038.png" /> there is a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274039.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274040.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032740/d03274041.png" />. In many categories, e.g. those of Abelian groups and modules over a ring, the categorical direct sum is given by the construction outlined above.
+
In category theory the direct sum or coproduct is defined by a universal property: Given objects $  X _ {i} $,  
 +
$  i \in I $,  
 +
in a category $  \mathfrak C $.  
 +
The direct sum $  Y = \oplus _ {i} X _ {i} $
 +
is an object of $  \mathfrak C $
 +
together with morphisms $  \alpha _ {i} : X _ {i} \rightarrow Y $
 +
such that for each object $  Z $
 +
of $  \mathfrak C $
 +
and family of morphisms $  \beta _ {i} : X _ {i} \rightarrow Z $
 +
there is a unique morphism $  \gamma : Y \rightarrow Z $
 +
such that $  \gamma \alpha _ {i} = \beta _ {i} $
 +
for all $  i \in I $.  
 +
In many categories, e.g. those of Abelian groups and modules over a ring, the categorical direct sum is given by the construction outlined above.
  
 
The direct sum is a special case of the [[restricted direct product]].
 
The direct sum is a special case of the [[restricted direct product]].

Latest revision as of 19:35, 5 June 2020


A construction extensively used in theories of mathematical structures which form categories similar to an Abelian category. In the non-Abelian case the direct sum is usually called the discrete direct product. Let $ \mathfrak A $ be a class of single-sorted algebraic systems which contain the one-element (zero) subsystems. The direct sum, or (discrete) direct product, of systems $ X _ {i} $, $ i \in I $, in $ \mathfrak A $ is the subsystem of the direct product $ X = \prod _ {i \in I } X _ {i} $ consisting of those functions $ f : I \rightarrow X $ for which all values, except for a finite number, belong to the corresponding zero subsystem. A direct sum is denoted by one of the following symbols:

$$ \prod _ {i \in I } {} ^ \otimes X _ {i} ,\ \ \prod _ {i \in I } {} ^ \oplus X _ {i} ,\ \ \sum _ {i \in I } X _ {i} ,\ \ \oplus _ {i \in I } X _ {i} $$

For a finite number of terms one also uses the notation:

$$ X _ {1} \dot{+} \dots \dot{+} X _ {n} . $$

The coincidence of the direct sum and the direct product in the case of a finite number of terms follows immediately from the definitions.

For each term of a direct sum $ X = \prod _ {i \in I } X _ {i} $ there exists a canonical imbedding $ q _ {i} : X _ {i} \rightarrow X $ that assigns to an element $ x \in X _ {i} $ the function $ q _ {i} ( x) : I \rightarrow X $, where $ q _ {i} ( x) $ takes the value $ x $ at the argument $ i $ and vanishes elsewhere. Hence it can be said that a direct sum contains its terms. In the case of an $ \Omega $- group (in particular, for groups, Abelian groups, vector spaces, and rings) one can give an "intrinsic" characterization of a direct sum. An $ \Omega $- group $ G $ is the direct sum of a family of $ \Omega $- subgroups $ G _ {i} $, $ i \in I $, if: a) $ G $ is generated by the $ G _ {i} $, $ i \in I $; b) each $ G _ {i} $ is an ideal in $ G $; and c) the intersection of $ G _ {i} $ with the $ \Omega $- subgroup generated by the remaining ideals is the trivial subgroup, for each $ i $. Cf. also Multi-operator group.

Every vector space is a direct sum of one-dimensional subspaces. Every free Abelian group is a direct sum of infinite cyclic groups. Every finite Abelian group is a direct sum of cyclic groups of prime-power order. Every semi-simple associative ring with a unit element and satisfying the minimum condition for ideals is the direct sum of a finite number of complete rings of linear transformations of appropriate finite-dimensional vector spaces.

In the theories of groups, lattices and categories, the isomorphism problem for direct decompositions has been extensively developed. Its origin is in the Remak–Schmidt theorem on central isomorphism of direct decompositions of groups having a principal series (cf. Krull–Remak–Schmidt theorem).

In category theory, the concept dual to that of a product, i.e. that of a coproduct, is sometimes called the direct sum.

Comments

As already noted the direct sum is also called the discrete direct product (cf. Direct product).

In category theory the direct sum or coproduct is defined by a universal property: Given objects $ X _ {i} $, $ i \in I $, in a category $ \mathfrak C $. The direct sum $ Y = \oplus _ {i} X _ {i} $ is an object of $ \mathfrak C $ together with morphisms $ \alpha _ {i} : X _ {i} \rightarrow Y $ such that for each object $ Z $ of $ \mathfrak C $ and family of morphisms $ \beta _ {i} : X _ {i} \rightarrow Z $ there is a unique morphism $ \gamma : Y \rightarrow Z $ such that $ \gamma \alpha _ {i} = \beta _ {i} $ for all $ i \in I $. In many categories, e.g. those of Abelian groups and modules over a ring, the categorical direct sum is given by the construction outlined above.

The direct sum is a special case of the restricted direct product.

How to Cite This Entry:
Direct sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Direct_sum&oldid=46711
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article