Concept
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. 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. 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. 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 . 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.