Zero-product property

For the product of zero factors, see empty product.

A zero product equation with three solutions.

In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, ${\displaystyle {\text{if }}ab=0,{\text{ then }}a=0{\text{ or }}b=0.}$

This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nonzero zero divisors , or one of the two zero-factor properties. ^[1] All of the number systems studied in elementary mathematics — the integers ${\displaystyle \mathbb {Z} }$, the rational numbers ${\displaystyle \mathbb {Q} }$, the real numbers ${\displaystyle \mathbb {R} }$, and the complex numbers ${\displaystyle \mathbb {C} }$— satisfy the zero-product property. In general, a ring which satisfies the zero-product property is called a domain.

Algebraic context

Suppose ${\displaystyle A}$ is an algebraic structure. We might ask, does ${\displaystyle A}$ have the zero-product property? In order for this question to have meaning, ${\displaystyle A}$ must have both additive structure and multiplicative structure. ^[2] Usually one assumes that ${\displaystyle A}$ is a ring, though it could be something else, e.g. the set of nonnegative integers ${\displaystyle \{0,1,2,\ldots \}}$ with ordinary addition and multiplication, which is only a (commutative) semiring.

Note that if ${\displaystyle A}$ satisfies the zero-product property, and if ${\displaystyle B}$ is a subset of ${\displaystyle A}$, then ${\displaystyle B}$ also satisfies the zero product property: if ${\displaystyle a}$ and ${\displaystyle b}$ are elements of ${\displaystyle B}$ such that ${\displaystyle ab=0}$, then either ${\displaystyle a=0}$ or ${\displaystyle b=0}$ because ${\displaystyle a}$ and ${\displaystyle b}$ can also be considered as elements of ${\displaystyle A}$.

Examples

• A ring in which the zero-product property holds is called a domain. A commutative domain is called an integral domain. Every field and every subring of a field are integral domains. Similarly, every subring of a division ring is a domain and satisfies the zero-product property.
• If ${\displaystyle p}$ is a prime number, then the ring of integers modulo ${\displaystyle p}$ 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.
• The zero-product property holds in the quaternions, since the quaternions form a division ring.
• The set of nonnegative integers ${\displaystyle \{0,1,2,\ldots \}}$ satisfies the zero-product property, as being a subset of the integers, which form an integral domain.

Non-examples

• Let ${\displaystyle \mathbb {Z} _{n}}$ denote the ring of integers modulo ${\displaystyle n}$. Then ${\displaystyle \mathbb {Z} _{6}}$ does not satisfy the zero product property: 2 and 3 are nonzero elements, yet ${\displaystyle 2\cdot 3\equiv 0{\pmod {6}}}$.
• In general, if ${\displaystyle n}$ is a composite number, then ${\displaystyle \mathbb {Z} _{n}}$ does not satisfy the zero-product property. Namely, if ${\displaystyle n=qm}$ where ${\displaystyle 0<q,m<n}$, then ${\displaystyle m}$ and ${\displaystyle q}$ are nonzero modulo ${\displaystyle n}$, yet ${\displaystyle qm\equiv 0{\pmod {n}}}$.
• The ring of 2×2 matrices with integer entries does not satisfy the zero-product property: if $${\displaystyle M={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}}$$ and ${\displaystyle N={\begin{pmatrix}0&1\\0&1\end{pmatrix}},}$ then $${\displaystyle MN={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}{\begin{pmatrix}0&1\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}=0,}$$ yet neither ${\displaystyle M}$ nor ${\displaystyle N}$ is zero.
• The ring of all functions ${\displaystyle f:[0,1]\to \mathbb {R} }$, 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 ${\displaystyle f_{1},\ldots ,f_{n}}$, none of which is identically zero, such that ${\displaystyle f_{i}\,f_{j}}$ is identically zero whenever ${\displaystyle i\neq j}$.
• 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 ${\displaystyle P}$ and ${\displaystyle Q}$ are univariate polynomials with real coefficients, and ${\displaystyle x}$ is a real number such that ${\displaystyle P(x)Q(x)=0}$. (Actually, we may allow the coefficients and ${\displaystyle x}$ to come from any integral domain.) By the zero-product property, it follows that either ${\displaystyle P(x)=0}$ or ${\displaystyle Q(x)=0}$. In other words, the roots of ${\displaystyle PQ}$ are precisely the roots of ${\displaystyle P}$ together with the roots of ${\displaystyle Q}$.

Thus, one can use factorization to find the roots of a polynomial. For example, the polynomial ${\displaystyle x^{3}-2x^{2}-5x+6}$ factorizes as ${\displaystyle (x-3)(x-1)(x+2)}$; hence, its roots are precisely 3, 1, and −2.

In general, suppose ${\displaystyle R}$ is an integral domain and ${\displaystyle f}$ is a monic univariate polynomial of degree ${\displaystyle d\geq 1}$ with coefficients in ${\displaystyle R}$. Suppose also that ${\displaystyle f}$ has ${\displaystyle d}$ distinct roots ${\displaystyle r_{1},\ldots ,r_{d}\in R}$. It follows (but we do not prove here) that ${\displaystyle f}$ factorizes as ${\displaystyle f(x)=(x-r_{1})\cdots (x-r_{d})}$. By the zero-product property, it follows that ${\displaystyle r_{1},\ldots ,r_{d}}$ are the only roots of ${\displaystyle f}$: any root of ${\displaystyle f}$ must be a root of ${\displaystyle (x-r_{i})}$ for some ${\displaystyle i}$. In particular, ${\displaystyle f}$ has at most ${\displaystyle d}$ distinct roots.

If however ${\displaystyle R}$ is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial ${\displaystyle x^{3}+3x^{2}+2x}$ has six roots in ${\displaystyle \mathbb {Z} _{6}}$ (though it has only three roots in ${\displaystyle \mathbb {Z} }$).

More properties

In a ring or, more generally, in a rng (ring without multiplicative identity) ⁠${\displaystyle R}$⁠, the multiplication has the (left) cancellation property if, for every nonzero element ⁠${\displaystyle a\in R}$⁠ one has $${\displaystyle ab=ac\implies b=c}$$ for every ⁠${\displaystyle b\in R}$⁠ and ⁠${\displaystyle c\in R}$⁠. The distributive property ⁠${\displaystyle a(b-c)=ab-ac}$⁠ implies that a ring or rng has the zero product property if and only if its multiplication has the left cancellation.

If ⁠${\displaystyle R}$⁠ is a commutative ring or a commutative rng with the zero-product property, one can define fractions similarly as for the definition of rational number. These fractions form a field called the field of fractions of ⁠${\displaystyle R}$⁠.

It follows that a commutative ring or rng has the zero-product property is and only if it is a subring or sub-rng of a field.

See also

• Fundamental theorem of algebra
• Prime ideal

Notes

1. 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.
2. 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

• PlanetMath: Zero rule of product