The product of two nonzero elements is nonzero
In
algebra, 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.
[1] 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 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.
[2] Usually one assumes that
is a
ring, 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. Any
field 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 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 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 ![{\displaystyle M={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/503427ce53b92344eedce4738a4b4454bf7a6e28)
and
then ![{\displaystyle MN={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}{\begin{pmatrix}0&1\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a75487d3d54d205b1b2bfb1b82a1277a8992b974)
yet neither
nor
is zero.
- The ring of all
functions
, 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 functions 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
Notes
-
^ The other being a⋅0 = 0⋅a = 0. Mustafa A. Munem and David J. Foulis, Algebra and Trigonometry with Applications (New York: Worth Publishers, 1982), p. 4.
-
^ There must be a notion of zero (the
additive identity) and a notion of products, i.e., multiplication.
References
- David S. Dummit and Richard M. Foote, Abstract Algebra (3d ed.), Wiley, 2003,
ISBN
0-471-43334-9.
External links