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On the Wikipedia page below, it says "a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units." https://en.wikipedia.org/wiki/Irreducible_element

On this Wolfram page it says, "An element $a$ of a ring which is nonzero, not a unit, and whose only divisors are the trivial ones."
http://mathworld.wolfram.com/IrreducibleElement.html

Can someone explain the discrepancy between these two definitions of Wikipedia and Wolfram? They are not the same definition, right? Is there actually a difference in definition of irreducible element of a general ring (i.e. no multiplicative identity, inverses, or commutativity) and that of a an integral domain? And, if so, what is the difference(s)? What is meant by "trivial ones", units?

I am being asked to explain how this question is different from a question asking about irreducible polynomials. Irreducible polynomials are not a part of this question, clearly..

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  • $\begingroup$ If by ring, you mean a commutative unital ring, then these are equivalent definitions. Wolfie's phrasing is a bit looser than Wikipedia's. $\endgroup$ – Lord Shark the Unknown Oct 27 '17 at 3:17
  • $\begingroup$ @LordSharktheUnknown That post is mine too. If it is a duplicate, can you explain how? $\endgroup$ – user3146 Oct 27 '17 at 3:39
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    $\begingroup$ @user3146 I think you should clarify yourself the question. Do you want to know if the two definitions are equivalent for integral domains? (Btw, for commutative rings in general there are some definitions of irreducible elements which are not equivalent, so these bear different names; see here.) $\endgroup$ – user26857 Oct 27 '17 at 14:31
  • $\begingroup$ @user26857 Thanks for the link! $\endgroup$ – user3146 Oct 27 '17 at 17:33
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    $\begingroup$ @user3146 Wolfram is wrong: the equivalence holds in integral domains and can fail in general (commutative rings). $\endgroup$ – user26857 Oct 27 '17 at 17:36

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