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If n is a positive integer and $_1, _2, _3, _4, \cdots,a_n$ are elements of an integral domain whose product is zero—i.e. $_1 * _2 * _3 * _4 ⋯ _ = 0$ — show that at least one of the $a’s$ is zero.

I was told to do this question by induction, however, I don’t see where I can apply that method here. Instead, I took the approach by using the definition of Integral domains. By definition, there are no proper divisors of zero, which implies that some $a$ multiplied with the product of the remaining a’s, which we denote by b which is the integral domain, equals zero where a = 0 or $b$ = 0. However I’m not sure if that’s sufficient enough to show that at least one of the a’s must be zero.

OVERTIME
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  • $a_1(a_2\ldotsa_n)=0$ implies $a_1=0$ or $a_2\ldotsa_n=0$. If $a_1=0$ we are done. In case that $a_2\ldotsa_n=0$ , by the same reasoning $a_2=0$ or $a_3\ldots*a_n=0$. Repeat inductively. – Deif Nov 20 '23 at 01:57
  • It's the same as the inductive proof of the $n$-ary version of Euclid's Lemma in the linked dupe (replace $,p\mid a,$ by $,a = 0,$ or, equivalently specialize $,p=0,,$ which is prime in a domain). – Bill Dubuque Nov 20 '23 at 02:00

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