# which powers of maximal ideal contain/are included. the notation

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)

• To be honest, I do not think that the second notion has a common used name.
– MooS
Commented Jul 17, 2015 at 6:18
• The second notion can be expressed via the Loewy length of the quotient, but I'd like some more direct/explicit notation. Commented Jul 18, 2015 at 5:27
• 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 $ll_R(I)$.
– user
Commented Jan 6, 2020 at 7:53