Zero-Product Property ○◂

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Zero-Product Property ○◂｜Definition｜1st｜20251119205401-00-⌔

Zero-product property - Wikipedia

Zero-product property

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

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 $\mathbb{Z}$, the rational numbers $\mathbb{Q}$, the real numbers $\mathbb{R}$, and the complex numbers $\mathbb{C}$ — satisfy the zero-product property. In general, a ring which satisfies the zero-product property is called a domain.

Printed 2026-06-28.

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Footnotes

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. ↩

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