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Zero-product property
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, it is the following assertion:
If , then or .
The zero-product property is also known as the rule of zero product, the null factor law or the nonexistence of nontrivial zero divisors. 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.〔There must be a notion of zero (the additive identity) and a notion of products, i.e., multiplication.〕 Usually one assumes that is a ring, though it could be something else, e.g., the nonnegative integers .
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 .
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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