Zero element

Inmathematics,azero elementis one of several generalizations ofthe number zeroto otheralgebraic structures.These alternate meanings may or may not reduce to the same thing, depending on the context.

Additive identities

Anadditive identityis theidentity elementin anadditive groupormonoid.It corresponds to the element 0 such that for all x in the group,0 +x=x+ 0 =x.Some examples of additive identity include:

Thezero vectorundervector addition:the vector whose components are all 0; in anormed vector spaceits norm (length) is also 0. Often denoted as $\mathbf {0}$ or ${\vec {0}}$ .^[1]
Thezero functionorzero mapdefined byz(x) = 0,underpointwise addition(f+g)(x) =f(x) +g(x)
Theempty setunderset union
Anempty sumoremptycoproduct
Aninitial objectin acategory(an empty coproduct, and so an identity undercoproducts)

Absorbing elements

Anabsorbing elementin a multiplicativesemigrouporsemiringgeneralises the property0 ⋅x= 0.Examples include:

Theempty set,which is an absorbing element underCartesian productof sets, since{ } ×S= { }
Thezero functionorzero mapdefined byz(x) = 0underpointwise multiplication(f⋅g)(x) =f(x) ⋅g(x)

Many absorbing elements are also additive identities, including the empty set and the zero function. Another important example is the distinguished element 0 in afieldorring,which is both the additive identity and the multiplicative absorbing element, and whoseprincipal idealis the smallest ideal.

Zero objects

Azero objectin acategoryis both aninitial and terminal object(and so an identity under bothcoproductsandproducts). For example, the trivial structure (containing only the identity) is a zero object in categories where morphisms must map identities to identities. Specific examples include:

Thetrivial group,containing only the identity (a zero object in thecategory of groups)
Thezero module,containing only the identity (a zero object in the category ofmodulesover a ring)

Zero morphisms

Azero morphismin acategoryis a generalised absorbing element underfunction composition:any morphism composed with a zero morphism gives a zero morphism. Specifically, if0_XY:X→Yis the zero morphism among morphisms fromXtoY,andf:A→Xandg:Y→Bare arbitrary morphisms, theng∘ 0_XY= 0_XBand0_XY∘f= 0_AY.

If a category has a zero object0,then there are canonical morphismsX→0and0→Y,and composing them gives a zero morphism0_XY:X→Y.In thecategory of groups,for example, zero morphisms are morphisms which always return group identities, thus generalising the functionz(x) = 0.

Least elements

Aleast elementin apartially ordered setorlatticemay sometimes be called a zero element, and written either as 0 or ⊥.

Zero module

Inmathematics,thezero moduleis themoduleconsisting of only the additiveidentityfor the module'sadditionfunction. In theintegers,this identity iszero,which gives the namezero module.That the zero module is in fact a module is simple to show; it is closed under addition andmultiplicationtrivially.

Zero ideal

Inmathematics,thezeroidealin aring $R$ is the ideal $\{0\}$ consisting of only the additive identity (orzeroelement). The fact that this is an ideal follows directly from the definition.

Zero matrix

Inmathematics,particularlylinear algebra,azero matrixis amatrixwith all its entries beingzero.It is alternately denoted by the symbol $O$ .^[2]Some examples of zero matrices are

0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}},\

The set ofm × nmatrices with entries in aringKforms a module $K_{m,n}$ .The zero matrix $0_{K_{m,n}}$ in $K_{m,n}$ is the matrix with all entries equal to $0_{K}$ ,where $0_{K}$ is the additive identity inK.

0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &&\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}

The zero matrix is the additive identity in $K_{m,n}$ .That is, for all $A\in K_{m,n}$ :

0_{K_{m,n}}+A=A+0_{K_{m,n}}=A

There is exactly one zero matrix of any given sizem × n(with entries from a given ring), so when the context is clear, one often refers tothezero matrix. In amatrix ring,the zero matrix serves the role of both an additive identity and an absorbing element. In general, the zero element of a ring is unique, and typically denoted as 0 without any subscript to indicate the parent ring. Hence the examples above represent zero matrices over any ring.

The zero matrix also represents thelinear transformationwhich sends all vectors to the zero vector.

Zero tensor

Inmathematics,thezero tensoris atensor,of any order, all of whose components arezero.The zero tensor of order 1 is sometimes known as the zero vector.

Taking atensor productof any tensor with any zero tensor results in another zero tensor. Among tensors of a given type, the zero tensor of that type serves as the additive identity among those tensors.

References

^Nair, M. Thamban; Singh, Arindama (2018).Linear Algebra.Springer. p. 3.doi:10.1007/978-981-13-0926-7.ISBN 978-981-13-0925-0.
^Lang, Serge(1987).Linear Algebra.Undergraduate Texts in Mathematics.Springer. p. 25.ISBN 9780387964126.We have a zero matrix in which $a_{ij}=0$ for all $i,j$ .... We shall write it $O$ .

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[1] Nair, M. Thamban; Singh, Arindama (2018).Linear Algebra.Springer. p. 3.doi:10.1007/978-981-13-0926-7.ISBN 978-981-13-0925-0.

[2] Lang, Serge(1987).Linear Algebra.Undergraduate Texts in Mathematics.Springer. p. 25.ISBN 9780387964126.We have a zero matrix in which $a_{ij}=0$ for all $i,j$ .... We shall write it $O$ .

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