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Hm, this would be good to be embedded more into other contexts. I have added a cross-link with group of units, but this would be good to expand on a little more.
in the mathematics literature a “multiplicative submonoid” as talked about in this article is called a multiplicative subset.
Thanks for chiming in. There is issues with what seems to be the same Anonymous editor also for instance here. If nobody speaks up to argue that there is value in these pages, we might just want to delete them.
multiplicative subset as described in the nlab article is a name for ’submonoid of the multiplicative monoid ’, rather than the more specific concept of ’the subset of cancellative elements in with respect to multiplication’. However, terminology in the nlab should be standardised, and so this article should be renamed multiplicative subset of cancellative elements, and a link established between the two articles.
However, I don’t think there is a single name universally accepted in the ring theory literature for the concept in this article. Lombardi and Quitté named this particular submonoid described in this article as the filter of regular elements on page 447 in Commutative algebra: Constructive methods (Finite projective modules) (regular element being a synonym of cancellative element used in the ring theory literature), and used it to define a unique factorization domain.
Just to say that it’s great to harmonize terminology across Lab entries where possible and sensible, but that the nature and sociology of this wiki makes a general standardization improbable if not impossible and hence not demandable. What really matters is that every single entry makes clear which of several imaginable variants it is talking about and what it’s context is, so that readers will be enabled to see for themselves.
With the present entry I am worried that it fails on this latter account. If you (Anonymous, Guest, anyone) see yourself adding worthwhile commentary and references as in #5 to the present entry, that would be welcome and might give this entry a reason to exist. Or else these comments might usefully go into other existing entries, and this one here be deleted, after all.
In my opinion this article is probably better off as an article about cancellative elements/regular elements in a commutative ring, rather than the subset of all cancellative elements/regular elements in a commutative ring. I’ve made the latter concept into a section of this article.
In fact, since the latter concept is defined as a subset in this article, one still needs to prove that the subset of cancellative elements forms a submonoid, i.e. that it contains and is closed under multiplication. I’ll get around to doing that some time later.
I’ve also added a proof that cancellative elements in a commutative ring are the same as non-zero-divisors as defined in the zero-divisor article on the nLab.
different anonymous person
fixing redirect for multiplicative submonoid of cancellative elements part 1
different anonymous person
multiplicative submonoid of cancellative elements and multiplicative subset of cancellative elements should now properly redirect to this article.
Redirect regular element in a ring which is not in a collision with regular element (of a Heyting algebra).
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