Namespaces
Variants
Actions

Difference between revisions of "Zero"

From Encyclopedia of Mathematics
Jump to: navigation, search
m
(rewording, links)
Line 4: Line 4:
 
$$0\cdot a = a \cdot 0 = 0 .$$
 
$$0\cdot a = a \cdot 0 = 0 .$$
  
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 zero of an Abelian group.
+
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 $A$ (in additive notation) is an element, also denoted by $0$, satisfying $0+a = a$ for all $a \in A$. It is uniquely determined.
+
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 $0$) has the property of absorption under 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 (cf. [[Zero divisor|Zero divisor]]). Fields, skew-fields and integral domains do not have zero divisors.
+
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 $0$) has the property of absorption under 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 $A$ (in multiplicative notation) is an 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.
+
A left zero of a [[semi-group]] $A$ (in multiplicative notation) is an 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.
+
The zero of a [[lattice]] is its 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.
+
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.
  
 
A zero is also called a null element.
 
A zero is also called a null element.

Revision as of 08:26, 28 December 2013


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 $0$) has the property of absorption under 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 an 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.

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=31216
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