In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, if a b = 0 , then a = 0 or b = 0. {\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. All of the number systems studied in elementary mathematics — the integers Z {\displaystyle \mathbb {Z} } , the rational numbers Q {\displaystyle \mathbb {Q} } , the real numbers R {\displaystyle \mathbb {R} } , and the complex numbers C {\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 A {\displaystyle A} is an algebraic structure. We might ask, does A {\displaystyle A} have the zero-product property? In order for this question to have meaning, A {\displaystyle A} must have both additive structure and multiplicative structure. Usually one assumes that A {\displaystyle A} is a ring, though it could be something else, e.g. the set of nonnegative integers { 0 , 1 , 2 , … } {\displaystyle \{0,1,2,\ldots \}} with ordinary addition and multiplication, which is only a (commutative) semiring. Note that if A {\displaystyle A} satisfies the zero-product property, and if B {\displaystyle B} is a subset of A {\displaystyle A} , then B {\displaystyle B} also satisfies the zero product property: if a {\displaystyle a} and b {\displaystyle b} are elements of B {\displaystyle B} such that a b = 0 {\displaystyle ab=0} , then either a = 0 {\displaystyle a=0} or b = 0 {\displaystyle b=0} because a {\displaystyle a} and b {\displaystyle b} can also be considered as elements of A {\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 p {\displaystyle p} is a prime number, then the ring of integers modulo p {\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 { 0 , 1 , 2 , … } {\displaystyle \{0,1,2,\ldots \}} satisfies the zero-product property, as being a subset of the integers, which form an integral domain.
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