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Let $(R,M)$ be a quasilocal ring which is not a field.

Can anyone explain this notation $(R,M)$?

Edit: The result is Let $(R,M)$ be a quasilocal ring which is not a field. If $M$ is nilpotent then $\Omega(R)$ is complete graph.

$\Omega(R)$ is graph whose vertices are from set of all nonzero annihilating ideals (denoted by $A(R)*$) of a commutative ring $R$ which is not integral domain and two distinct vertices $I$ and $J$ are joined by an edge if $I+J\in A(R).$ ($A(R)$ is set of annihilating ideals of $R$)

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  • $\begingroup$ Perhaps $M$ is the finite set of all maximal ideals.. Can you give more context, e.g. how is $M$ used in the text? $\endgroup$
    – Berci
    Commented Jan 5, 2021 at 8:38
  • $\begingroup$ Okay i will add more information about it in question. $\endgroup$
    – sushil soni
    Commented Jan 5, 2021 at 8:41
  • $\begingroup$ Do you mind adding a link to the book/paper where you've found this? I think more context might still be helpful $\endgroup$
    – Chris Grossack
    Commented Jan 5, 2021 at 8:52
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    $\begingroup$ I know some authors use "quasilocal" to mean a non-noetherian ring with a unique maximal ideal. In that case, $M$ is likely this ideal $\endgroup$
    – Chris Grossack
    Commented Jan 5, 2021 at 8:53
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    $\begingroup$ From the paper it looks like $M$ is indeed the unique maximal ideal of $R$. That makes sense contextually, and agrees with a definition of quasi-local that I've seen before. That said, I'm not familiar enough with this material to feel confident leaving that as an answer. $\endgroup$
    – Chris Grossack
    Commented Jan 5, 2021 at 9:11

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The notation for "a quasilocal ring $(R,M)$" is shorthand for "a ring $R$ with a unique maximal ideal $M$."

It's similar to one notation used to specify a ring by elaborating on all of the data required to define it, like "$(R,+,\cdot)$".

It's worth mentioning that the usage of "quasilocal" has been declining since the 1960's and now will typically just appear as "local". The notation $(R,M)$ has persisted through the change, apparently.

In the course of writing the answer and participating in the comments, some users mentioned the possibility of using quasilocal to mean "finitely many maximal ideals."

I can say with some confidence that (in English) I have never seen that usage in a book or in an article. The term that does mean that for commutative rings, and which appears consistently is semilocal.

In a google search for "quasilocal ring" there were hits from Wolfram Mathworld and CommAlg which used the "multiple maximal ideals" version. However, I've found these two resources are not reliable. By that I mean this is not the first time I've found them to be inconsistent with the rest of algebra literature. I could not find any usages of this sort in google books. On the other hand, the Wikipedia support the usages in the way I'd expect.

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    $\begingroup$ It's worth mentioning that this term is confusing because "quasilocal" is sometimes used to mean a (commutative) ring with finitely many maximal ideals (this is, for example, the first and second Google hit for me when searching "quasilocal ring"). I gather here that in the convention where "quasilocal" means "unique maximal ideal," "local" means "Noetherian and has a unique maximal ideal"? $\endgroup$
    – Qiaochu Yuan
    Commented Jan 5, 2021 at 20:07
  • $\begingroup$ @QiaochuYuan Does that usage even deserve any credence though? I'm sure my hits aren't exactly the same as yours, but in my list I do see hits from Wolfram and CommAlg which are (in my experience anyway) unreliable resources. They are, unfortunately, pushed to the top of the search. I couldn't find any usages of that sort in google books, when I did the same search, but I only checked the first three pages or so. I'm not aware of any text in English using the term that way... are you? $\endgroup$
    – rschwieb
    Commented Jan 5, 2021 at 20:30
  • $\begingroup$ Nope, but I don't read commutative algebra texts. I guess the common term for "finitely many maximal ideals" is "semilocal" (which also has a MathWorld article, but maybe more importantly has a Wikipedia article), but I can't speak with any authority about that. $\endgroup$
    – Qiaochu Yuan
    Commented Jan 5, 2021 at 20:34
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    $\begingroup$ @QiaochuYuan Yes, the term semilocal is very established. $\endgroup$
    – rschwieb
    Commented Jan 5, 2021 at 20:39

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