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Results on Dilworth and Rees numbers of Artinian local rings. (English) Zbl 0857.13014
Let \(A\) be an Artinian local ring with \(m\) its maximal ideal. J. Watanabe introduced the Dilworth number \(d(A)=\max \{\mu(I) \mid I\) is an ideal of \(A\}\) and the Rees number \(r(A)=\min \{\ell(A/xA) \mid x\in m\}\) (here \(\mu(I)\) is the minimal number of generators of the ideal \(I\) and \(\ell(M)\) is the length of the \(A\)-module \(M)\), and showed that \(d(A)\leq r(A)\). This paper contains several results on \(d(A)\) and \(r(A)\). For example, the author shows that \(d(A)<d(B) \leq 2d(A)\) and \(r(A)<r(B) \leq 2r(A)\) for \(B=A[T]/(T^2)\), the trivial deformation of \(A\), and investigates \(d(A)\) and \(r(A)\) when \(A\) is an Artinian graded ring.

13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13E15 Commutative rings and modules of finite generation or presentation; number of generators