Let $R$ be a (associative, commutative) local ring, denote by $\mathfrak{m}$ its maximal ideal. For any other ideal $J\subset R$ one can speak about:

  • the biggest power $k\le\infty$ such that $J\subseteq\mathfrak{m}^k$
  • the smallest power k such that $J\supseteq\mathfrak{m}^k$ (assuming $J$ contains some power of the maximal ideal)

What are the standard notations for these numbers? (for the first number I'd use $ord_{\mathfrak{m}}(J)$, but no idea about the second number)

  • $\begingroup$ To be honest, I do not think that the second notion has a common used name. $\endgroup$
    – MooS
    Commented Jul 17, 2015 at 6:18
  • $\begingroup$ The second notion can be expressed via the Loewy length of the quotient, but I'd like some more direct/explicit notation. $\endgroup$ Commented Jul 18, 2015 at 5:27
  • $\begingroup$ In case you're still interested, the usual notation for the Loewy length of an $\mathfrak m$-primary ideal $I$ in a Noetherian local ring $(R, \mathfrak m)$ is $ll_R(I)$. $\endgroup$
    – user
    Commented Jan 6, 2020 at 7:53


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