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The Dilworth number of Artinian rings and finite posets with rank function. (English) Zbl 0648.13010
Commutative algebra and combinatorics, US-Jap. Joint Semin., Kyoto/Jap. 1985, Adv. Stud. Pure Math. 11, 303-312 (1987).
[For the entire collection see Zbl 0632.00003.]
In an earlier paper, the author proved that if A is an Artinian local ring, \({\mathfrak a}\) an arbitrary ideal of A and y an arbitrary non-unit element of A, then \(\mu({\mathfrak a})\), the minimal number of generators of A, is less than or equal to \(\ell (A/yA)\), the length of \(A/yA\). The present paper gives a combinatorial interpretation of the number \(d(A)=Max\{\mu ({\mathfrak a})| {\mathfrak a}\) an ideal of \(A\}\) and studies one case where \(d(A)=Min\{\ell (A/yA)| y\) a non-unit of \(A\}\).
Reviewer: W.M.Cunnea

MSC:
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13E15 Commutative rings and modules of finite generation or presentation; number of generators
13H99 Local rings and semilocal rings