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."
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..