Gorenstein projective and injective dimensions over Frobenius extensions

Let $R\subset A$ be a Frobenius extension of rings. We prove that: (1) for any left $A$-module $M$, $_{A}M$ is Gorenstein projective (injective) if and only if the underlying left $R$-module $_{R}M$ is Gorenstein projective (injective). (2) if $\mathrm{G}\text{-}\mathrm{proj.dim}_{A}M<\infty$, then $\mathrm{G}\text{-}\mathrm{proj.dim}_{A}M = \mathrm{G}\text{-}\mathrm{proj.dim}_{R}M$, the dual for Gorenstein injective dimension also holds. (3) if the extension is split, then $\mathrm{G}\text{-}\mathrm{gldim}(A)= \mathrm{G}\text{-}\mathrm{gldim}(R)$.Comment: A corrigendum version of Comm. Algebra,46(12):5348-5354, 2018. A typo in Proposition 3.2 is fixed, and the assumption that the extension is split is added for Theorem 3.3, 3.4, and Corollary 3.5. arXiv admin note: text overlap with arXiv:1707.0588

A Note on: `Algorithms for Connected Set Cover Problem and Fault-Tolerant Connected Set Cover Problem'

A flaw in the greedy approximation algorithm proposed by Zhang et al. for minimum connected set cover problem is corrected, and a stronger result on the approximation ratio of the modified greedy algorithm is established. The results are now consistent with the existing results on connected dominating set problem which is a special case of the minimum connected set cover problem.Comment: 6 pages, 1 figure, submitted to Theoretical Computer Scienc