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A (real or complex) number having the property that any number does not change if zero is added to it. It is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z0992001.png" />. The product of any number with zero is zero:
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{{TEX|done}}
  
<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/z/z099/z099200/z0992002.png" /></td> </tr></table>
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A (real or complex) number having the property that any number does not change if zero is added to it. It is denoted by the symbol $0$. The product of any number with zero is zero:
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$$0\cdot a = a \cdot 0 = 0 .$$
  
If the product of two numbers is zero, then one of them is zero (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z0992003.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z0992004.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z0992005.png" />). Division by zero is not defined. A direct generalization of this concept is that of the zero of an Abelian group.
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If the product of two numbers is zero, then one of them is zero (that is, $a\cdot b = 0$ implies $a=0$ or $b=0$). Division by zero is not defined.  
  
The zero of an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z0992006.png" /> (in additive notation) is an element, also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z0992007.png" />, satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z0992008.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z0992009.png" />. It is uniquely determined.
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A direct generalization of this concept is that of the identity element of an [[Abelian group]]: the zero of an Abelian group $A$ (in additive notation) is an element, also denoted by $0$, satisfying $0+a = a$ for all $a \in A$. It is uniquely determined.
  
The zero of a ring (in particular, of a skew-field, i.e. division ring, or a field) is the zero of its additive group. The zero of a ring (like the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920010.png" />) has the property of absorption under multiplication: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920011.png" />. However, in an arbitrary ring the product of two non-zero elements may be zero. Such elements are called zero divisors (cf. [[Zero divisor|Zero divisor]]). Fields, skew-fields and integral domains do not have zero divisors.
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The zero of a [[ring]] (in particular, of a [[skew-field]], i.e. [[division ring]], or a [[field]]) is the zero of its additive group. The zero of a ring (like the number $0$) is an [[absorbing element]] for multiplication: $0\cdot a = a \cdot 0 = 0$. However, in an arbitrary ring the product of two non-zero elements may be zero. Such elements are called [[zero divisor]]s. Fields, skew-fields and [[integral domain]]s do not have zero divisors.
  
A left zero of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920012.png" /> (in multiplicative notation) is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920015.png" />. A right zero is defined by the dual property. If a semi-group has a two-sided zero (an element which is both a left and a right zero), then this element is unique. The zero of a ring is also the zero of its multiplicative semi-group.
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A left zero of a [[semi-group]] $A$ (in multiplicative notation) is a left absorbing element $0 \in A$ such that $0\cdot a = 0$ for all $a \in A$. A right zero is defined by the dual property. If a semi-group has a two-sided zero (an element which is both a left and a right zero), then this element is unique. The zero of a ring is also the zero of its multiplicative semi-group.
  
The zero of a lattice is its minimal element, if this exists. A complete lattice always has a zero: the intersection of all elements.
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The zero of a [[partially ordered set]], such as a [[lattice]], is a minimal element, if this exists. A [[complete lattice]] always has a zero: the intersection of all elements.
  
A zero of an algebraic system is an element picked out by a nullary operation (see [[Algebraic operation|Algebraic operation]]; [[Algebraic system|Algebraic system]]). In the majority of examples considered above the zero is unique in the given system and even forms a one-element subsystem.
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A zero of an algebraic system is an element picked out by a [[nullary operation]] (see [[Algebraic operation]]; [[Algebraic system]]). In the majority of examples considered above the zero is unique in the given system and even forms a one-element subsystem.
  
 
A zero is also called a null element.
 
A zero is also called a null element.
  
For a zero object of a category, see [[Null object of a category|Null object of a category]].
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For a zero object of a category, see [[Null object of a category]].
 
 
The set of zeros of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920016.png" /> taking values in an Abelian group (ring, field, skew-field) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920017.png" /> is the collection of values of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920018.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920019.png" />.
 
  
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The set of zeros of a function $f(x_1,\ldots,x_n)$ taking values in an Abelian group (ring, field, skew-field) $A$ is the collection of values of the variables $(x_1,\ldots,x_n)$ for which $f(x_1,\ldots,x_n) = 0$.
  
  
 
====Comments====
 
====Comments====
A subset of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920020.png" /> is called a zero set if it is the set of zeros of some continuous real-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099200/z09920021.png" />. Zero sets are an object of study in algebraic geometry (zero sets of systems of polynomials) and local analytic geometry (zero sets of systems of holomorphic functions and mappings).
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A subset of a topological space $X$ is called a zero set if it is the set of zeros of some continuous real-valued function on $X$. Zero sets are an object of study in algebraic geometry (zero sets of systems of polynomials) and local analytic geometry (zero sets of systems of holomorphic functions and mappings).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Jacobson,   "Basic algebra" , '''1''' , Freeman (1974)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Jacobson, "Basic algebra" , '''1''' , Freeman (1974) {{MR|0356989}} {{ZBL|0284.16001}} </TD></TR></table>

Latest revision as of 21:06, 3 January 2015


A (real or complex) number having the property that any number does not change if zero is added to it. It is denoted by the symbol $0$. The product of any number with zero is zero: $$0\cdot a = a \cdot 0 = 0 .$$

If the product of two numbers is zero, then one of them is zero (that is, $a\cdot b = 0$ implies $a=0$ or $b=0$). Division by zero is not defined.

A direct generalization of this concept is that of the identity element of an Abelian group: the zero of an Abelian group $A$ (in additive notation) is an element, also denoted by $0$, satisfying $0+a = a$ for all $a \in A$. It is uniquely determined.

The zero of a ring (in particular, of a skew-field, i.e. division ring, or a field) is the zero of its additive group. The zero of a ring (like the number $0$) is an absorbing element for multiplication: $0\cdot a = a \cdot 0 = 0$. However, in an arbitrary ring the product of two non-zero elements may be zero. Such elements are called zero divisors. Fields, skew-fields and integral domains do not have zero divisors.

A left zero of a semi-group $A$ (in multiplicative notation) is a left absorbing element $0 \in A$ such that $0\cdot a = 0$ for all $a \in A$. A right zero is defined by the dual property. If a semi-group has a two-sided zero (an element which is both a left and a right zero), then this element is unique. The zero of a ring is also the zero of its multiplicative semi-group.

The zero of a partially ordered set, such as a lattice, is a minimal element, if this exists. A complete lattice always has a zero: the intersection of all elements.

A zero of an algebraic system is an element picked out by a nullary operation (see Algebraic operation; Algebraic system). In the majority of examples considered above the zero is unique in the given system and even forms a one-element subsystem.

A zero is also called a null element.

For a zero object of a category, see Null object of a category.

The set of zeros of a function $f(x_1,\ldots,x_n)$ taking values in an Abelian group (ring, field, skew-field) $A$ is the collection of values of the variables $(x_1,\ldots,x_n)$ for which $f(x_1,\ldots,x_n) = 0$.


Comments

A subset of a topological space $X$ is called a zero set if it is the set of zeros of some continuous real-valued function on $X$. Zero sets are an object of study in algebraic geometry (zero sets of systems of polynomials) and local analytic geometry (zero sets of systems of holomorphic functions and mappings).

References

[a1] N. Jacobson, "Basic algebra" , 1 , Freeman (1974) MR0356989 Zbl 0284.16001
How to Cite This Entry:
Zero. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zero&oldid=17671
This article was adapted from an original article by O.A. IvanovaL.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article