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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.

MSC:
 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