= Integral domain
{title2=$ab=0\Rightarrow a=0\text{ or }b=0$}
{wiki}
An <integral domain> is a nonzero <commutative ring> without <zero divisors>. Equivalently, a product of two nonzero elements is always nonzero. The <zero ring> is excluded, and $\mathbb Z/6\mathbb Z$ fails because $2\cdot3=0$. Every <field> is an <integral domain>, but the <integers> show that the converse fails: $2$ has no <multiplicative inverse> in $\mathbb Z$. See https://www.ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/0c771e6ff658b800f22aa90016880028_MIT18_703S13_pra_l_15.pdf[MIT's Basic Properties of Rings, definitions 15.9–15.10 and example 15.11].
The practical benefit is <multiplicative cancellation>. In the <integers>, $2x=6$ has at most one solution and indeed $x=3$ works. Yet $2x=1$ has no <integer> solution. Thus <multiplicative cancellation> guarantees uniqueness when a solution exists; it does not guarantee division by every nonzero element. Passing to the <fraction field> $\mathbb Q$ supplies the missing solution $1/2$.
The <polynomial ring> $\mathbb Z[X]$ is also an <integral domain>. For example, the product of $2X+1$ and $3X-4$ has <leading coefficient> $6$ and cannot vanish. Generally the <leading coefficient> of a product is the product of the two <leading coefficients>; an <integral domain> prevents that coefficient from vanishing. This connects <integer> arithmetic to <polynomial> arithmetic through one reusable principle.
Historically, this connection grew far beyond elementary examples. Keith Conrad identifies the <ring of integers> of a <number field> and a <ring> of <polynomial functions> on a smooth <plane curve> as early important examples of <Dedekind domains>. This shared structure connects <number theory> with <algebraic geometry>; see https://kconrad.math.uconn.edu/blurbs/gradnumthy/idealfactor.pdf[Conrad's Ideal Factorization, section 6].
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