Numbers of generators of ideals in a group ring of an elementary Abelian \(p\)-group
Let \(\mu(I)\) denote the minimal number of generators of an ideal \(I\) of a local ring \((R,M)\). The numbers \(d(R) = \max \{ \mu(I) :I\text{ is an ideal of } R \}\) and \( \operatorname {sp}(R) = \max \{ \mu (M^{i}): i \geq 0 \}\) are called, respectively, the Dilworth and the Sperner number of \(R\). The paper is mainly devoted to prove that if \((A,m)\) is an Artin local ring of characteristic \(p^{n}\), \(n > 1\), \(m = zA\) with \(pA = z^{e}A\), \(G=(\mathbb{Z}/p\mathbb{Z})^{k}\), \(\alpha = \min \{e, \varphi(p)\}\) and \(m^{(k-1) \varphi(p) + \alpha} \neq 0\), then the Dilworth and Sperner numbers of the group ring \(A[G]\) coincide and they are equal to \(1 + \frac{\alpha}{\varphi(p)} (|G|-1)\), where \(\varphi\) denotes the Euler function. Moreover, the authors show that whenever \((R,m)\) is a \(1\)-dimensional local Noetherian ring with \(R/m\) of characteristic \(p\) and \(G\) is a finite Abelian \(p\)-group, then the Dilworth numbers of \(R[G]\) and of \((R/m^{c})[G]\) coincide for \(c\) large, and so it happens for the Sperner numbers. The authors apply this to obtain the Dilworth and Sperner numbers of the group ring \(A[G]\) for certain one-dimensional group rings.
scientific article; zbMATH DE number 1423892
scientific article