Direct sum

The direct sum is an operation from abstract algebra (a branch of mathematics). For example, the direct sum R ⊕ R {displaystyle mathbf {R} oplus mathbf {R} } , where R {displaystyle mathbf {R} } is real coordinate space, is the Cartesian plane, R 2 {displaystyle mathbf {R} ^{2}} . To see how direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups A {displaystyle A} and B {displaystyle B} is another abelian group A ⊕ B {displaystyle Aoplus B} consisting of the ordered pairs ( a , b ) {displaystyle (a,b)} where a ∈ A {displaystyle ain A} and b ∈ B {displaystyle bin B} . (Confusingly this ordered pair is also called the cartesian product of the two groups.) To add ordered pairs, we define the sum ( a , b ) + ( c , d ) {displaystyle (a,b)+(c,d)} to be ( a + c , b + d ) {displaystyle (a+c,b+d)} ; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of any two algebraic structures, such as rings, modules, and vector spaces. We can also form direct sums with any finite number of summands, for example A ⊕ B ⊕ C {displaystyle Aoplus Boplus C} , provided A , B , {displaystyle A,B,} and C {displaystyle C} are the same kinds of algebraic structures (that is, all groups, rings, vector spaces, etc.). This relies on the fact that the direct sum is associative up to isomorphism. That is, ( A ⊕ B ) ⊕ C ≅ A ⊕ ( B ⊕ C ) {displaystyle (Aoplus B)oplus Ccong Aoplus (Boplus C)} for any algebraic structures A {displaystyle A} , B {displaystyle B} , and C {displaystyle C} of the same kind. The direct sum is also commutative up to isomorphism, i.e. A ⊕ B ≅ B ⊕ A {displaystyle Aoplus Bcong Boplus A} for any algebraic structures A {displaystyle A} and B {displaystyle B} of the same kind. In the case of two summands, or any finite number of summands, the direct sum is the same as the direct product. If the arithmetic operation is written as +, as it usually is in abelian groups, then we use the direct sum. If the arithmetic operation is written as × or ⋅ or using juxtaposition (as in the expression x y {displaystyle xy} ) we use direct product. In the case where infinitely many objects are combined, most authors make a distinction between direct sum and direct product. As an example, consider the direct sum and direct product of infinitely many real lines. An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there would be a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both. More generally, if a + sign is used, all but finitely many coordinates must be zero, while if some form of multiplication is used, all but finitely many coordinates must be 1. In more technical language, if the summands are ( A i ) i ∈ I {displaystyle (A_{i})_{iin I}} , the direct sum ⨁ i ∈ I A i {displaystyle igoplus _{iin I}A_{i}} is defined to be the set of tuples ( a i ) i ∈ I {displaystyle (a_{i})_{iin I}} with a i ∈ A i {displaystyle a_{i}in A_{i}} such that a i = 0 {displaystyle a_{i}=0} for all but finitely many i. The direct sum ⨁ i ∈ I A i {displaystyle igoplus _{iin I}A_{i}} is contained in the direct product ∏ i ∈ I A i {displaystyle prod _{iin I}A_{i}} , but is usually strictly smaller when the index set I {displaystyle I} is infinite, because direct products do not have the restriction that all but finitely many coordinates must be zero. The xy-plane, a two-dimensional vector space, can be thought of as the direct sum of two one-dimensional vector spaces, namely the x and y axes. In this direct sum, the x and y axes intersect only at the origin (the zero vector). Addition is defined coordinate-wise, that is ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) {displaystyle (x_{1},y_{1})+(x_{2},y_{2})=(x_{1}+x_{2},y_{1}+y_{2})} , which is the same as vector addition. Given two structures A {displaystyle A} and B {displaystyle B} , their direct sum is written as A ⊕ B {displaystyle Aoplus B} . Given an indexed family of structures A i {displaystyle A_{i}} , indexed with i ∈ I {displaystyle iin I} , the direct sum may be written A = ⨁ i ∈ I A i {displaystyle extstyle A=igoplus _{iin I}A_{i}} . Each Ai is called a direct summand of A. If the index set is finite, the direct sum is the same as the direct product. In the case of groups, if the group operation is written as + {displaystyle +} the phrase 'direct sum' is used, while if the group operation is written ∗ {displaystyle *} the phrase 'direct product' is used. When the index set is infinite, the direct sum is not the same as the direct product since the direct sum has the extra requirement that all but finitely many coordinates must be zero. A distinction is made between internal and external direct sums, though the two are isomorphic. If the factors are defined first, and then the direct sum is defined in terms of the factors, we have an external direct sum. For example, if we define the real numbers R {displaystyle mathbf {R} } and then define R ⊕ R {displaystyle mathbf {R} oplus mathbf {R} } the direct sum is said to be external. If, on the other hand, we first define some algebraic structure S {displaystyle S} and then write S {displaystyle S} as a direct sum of two substructures V {displaystyle V} and W {displaystyle W} , then the direct sum is said to be internal. In this case, each element of S {displaystyle S} is expressible uniquely as an algebraic combination of an element of V {displaystyle V} and an element of W {displaystyle W} . For an example of an internal direct sum, consider Z 6 {displaystyle mathbb {Z} _{6}} (the integers modulo six), whose elements are { 0 , 1 , 2 , 3 , 4 , 5 } {displaystyle {0,1,2,3,4,5}} . This is expressible as an internal direct sum Z 6 = { 0 , 2 , 4 } ⊕ { 0 , 3 } {displaystyle mathbb {Z} _{6}={0,2,4}oplus {0,3}} .