In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, the zero-product property states that the product of two
nonzero elements is nonzero. In other words,
This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial
zero divisors, or one of the two zero-factor properties. All of the
number systems studied in
elementary mathematics — the
integers
, the
rational numbers
, the
real numbers
, and the
complex numbers
— satisfy the zero-product property. In general, a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
which satisfies the zero-product property is called a
domain.
Algebraic context
Suppose
is an algebraic structure. We might ask, does
have the zero-product property? In order for this question to have meaning,
must have both additive structure and multiplicative structure.
[There must be a notion of zero (the ]additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from elem ...
) and a notion of products, i.e., multiplication. Usually one assumes that
is a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, though it could be something else, e.g. the set of nonnegative integers
with ordinary addition and multiplication, which is only a (commutative)
semiring.
Note that if
satisfies the zero-product property, and if
is a subset of
, then
also satisfies the zero product property: if
and
are elements of
such that
, then either
or
because
and
can also be considered as elements of
.
Examples
* A ring in which the zero-product property holds is called a
domain. A
commutative domain with a
multiplicative identity element is called an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
. Any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
is an integral domain; in fact, any subring of a field is an integral domain (as long as it contains 1). Similarly, any subring of a
skew field
Skew may refer to:
In mathematics
* Skew lines, neither parallel nor intersecting.
* Skew normal distribution, a probability distribution
* Skew field or division ring
* Skew-Hermitian matrix
* Skew lattice
* Skew polygon, whose vertices do not ...
is a domain. Thus, the zero-product property holds for any subring of a skew field.
* If
is a
prime number, then the ring of
integers modulo has the zero-product property (in fact, it is a field).
* The
Gaussian integers are an
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
because they are a subring of the complex numbers.
* In the
strictly skew field of
quaternions, the zero-product property holds. This ring is not an integral domain, because the multiplication is not commutative.
* The set of nonnegative integers
is not a ring (being instead a
semiring), but it does satisfy the zero-product property.
Non-examples
* Let
denote the ring of
integers modulo . Then
does not satisfy the zero product property: 2 and 3 are nonzero elements, yet
.
* In general, if
is a
composite number, then
does not satisfy the zero-product property. Namely, if
where
, then
and
are nonzero modulo
, yet
.
* The ring
of 2×2
matrices with
integer entries does not satisfy the zero-product property: if
and
then
yet neither
nor
is zero.
* The ring of all
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s
, from the
unit interval to the
real numbers, has nontrivial zero divisors: there are pairs of functions which are not identically equal to zero yet whose product is the zero function. In fact, it is not hard to construct, for any ''n'' ≥ 2, functions
, none of which is identically zero, such that
is identically zero whenever
.
* The same is true even if we consider only continuous functions, or only even infinitely
smooth functions. On the other hand,
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s have the zero-product property.
Application to finding roots of polynomials
Suppose
and
are univariate polynomials with real coefficients, and
is a real number such that
. (Actually, we may allow the coefficients and
to come from any integral domain.) By the zero-product property, it follows that either
or
. In other words, the roots of
are precisely the roots of
together with the roots of
.
Thus, one can use
factorization to find the roots of a polynomial. For example, the polynomial
factorizes as
; hence, its roots are precisely 3, 1, and −2.
In general, suppose
is an integral domain and
is a
monic univariate polynomial of degree
with coefficients in
. Suppose also that
has
distinct roots
. It follows (but we do not prove here) that
factorizes as
. By the zero-product property, it follows that
are the ''only'' roots of
: any root of
must be a root of
for some
. In particular,
has at most
distinct roots.
If however
is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial
has six roots in
(though it has only three roots in
).
See also
*
Fundamental theorem of algebra
*
Integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
and
domain
*
Prime ideal
*
Zero divisor
Notes
References
*David S. Dummit and Richard M. Foote, ''Abstract Algebra'' (3d ed.), Wiley, 2003, {{isbn, 0-471-43334-9.
External links
PlanetMath: Zero rule of product
Abstract algebra
Elementary algebra
Real analysis
Ring theory
0 (number)