4 ms·
Well, if you're willing to allow a bit of wiggle room in the form of "up to associates" (which is a qualifier that I forgot to add), there are integral domains
by jackmaney 12y ago
Well, if you're willing to allow a bit of wiggle room in the form of "up to associates" (which is a qualifier that I forgot to add), there are integral domains in which every nonzero element equals the product of a unit and a power of a fixed nonzero nonunit element. This qualifier isn't too much to ask since "up to associates" is the generalized version of "up to a plus or minus" when factoring integers (which has no real bearing on factorizations).
In particular, let F be any field, and consider F[[x]], the ring of formal power series[1] over F. Given any nonzero f(x) in F[[x]], we can "factor out as many x's as we can" to write f(x) = x^n g(x) where n is a nonnegative integer, g(x) is in F[[x]], and the constant term of g(x) is nonzero (here we explicitly define x^0 to be 1). Thus, g(x) is a unit[2] (ie invertible), and every nonzero element of F[[x]] is a unit multiple of a power of x.
[1]: http://en.wikipedia.org/wiki/Formal_power_series http://en.wikipedia.org/wiki/Formal_power_series
[2]: http://en.wikipedia.org/wiki/Formal_power_series#Inverting_series http://en.wikipedia.org/wiki/Formal_power_series#Inverting_s...
- tprice7 12y agoOk, but rings for which every nonzero element is a power of a fixed nonzero nonunit element up to associates are exactly discrete valuation rings. So your statement boils down to "The only UFDs that are discrete valuation rings are (rank 1) discrete valuation rings". And I'm not sure what you mean by rank 1 here, but I can deduce that all discrete valuation rings are rank 1 from what you've said, since all discrete valuation rings are UFDs. So what you've said boils down to: "The only UFDs that are discrete valuation rings are discrete valuation rings", which again doesn't make UFDs more relevant than any other condition you might add.
- tprice7 12y agoCorrection: the _integral domains_ for which every nonzero element is a power of a fixed nonzero nonunit element up to associates are exactly discrete valuation rings. So my final statement should read: "...which again doesn't make UFDs more relevant than any other condition you might add _that is stronger than the much more general condition of being an integral domain_"
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