In my book, UFD is defined as - if every nonzero nonunit element can be uniquely written as factors of irreducible elements. But wiki says non-zero element can be written as product of factors of irredicible elements and unit (https://en.wikipedia.org/wiki/Unique_factorization_domain). Pls enlighten on how these 2 are equivalent.

  • $\begingroup$ The issue is uniqueness. If we regard associated irreducible elements as not "distinct", then the definitions are equivalent. The unit factors can throw off the uniqueness, unless the definition of "distinct" irreducibles makes associated irreducibles non-distinct. $\endgroup$ – quasi Oct 16 '17 at 18:57

Your book should really add the phrase "up to units" as part of the definition. Otherwise the following is technically not unique $6=2\cdot 3 = (-2)\cdot (-3) $ but $\mathbb Z$ is the mother of all UFDs.

Another way I've seen this dealt with is the notion of associates. Two non-zero elements $a,b$ in a ring are said to be associate if there is a unit $u$ such that $a=bu$. My favourite statement of unique factorization domain is a domain such that every non-zero element is expressible as a product of irreducible elements (which have to be non-units) and any two such expressions $a_1\cdots a_n$ = $b_1\cdots b_m$ means that $n=m$ and there is some reordering so that $a_i$ and $b_i$ are associates.

  • $\begingroup$ let me understand slowly. I am novice trying to learn ring theory. "upto units" means product till unit ie., 6 = 2.3.1 ( 1 - unit). i think this ensures uniqueness i.e, removes factorisation (-2)*(-3). Pls correct if i am wrong $\endgroup$ – Magneto Oct 16 '17 at 19:14
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    $\begingroup$ It doesn't remove the factorization per say, it just says that it's the same as 2*3 because it only differs by the units -1 $\endgroup$ – RKD Oct 16 '17 at 19:18

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