Let $R$ be a commutative Noetherian ring. It is shown that the finitely generated $R$-module $M$ with finite Gorenstein dimension is reflexive if and only if $M_{𝔭}$ is reflexive for $𝔭\in \mathrm{Spec}\left(R\right)$ with $\mathrm{depth}\left(R_{𝔭}\right)\le 1$, and ${\text{G-}\text{dim}}_{R_{𝔭}}\left(M_{𝔭}\right)\le \mathrm{depth}\left(R_{𝔭}\right)-2$ for $𝔭\in \mathrm{Spec}\left(R\right)$ with $\mathrm{depth}\left(R_{𝔭}\right)\ge 2$. This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for $n\ge 2$ we give a characterization of $n$-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown...

We give a short overview on the subject of canonical reduction of a pair of bilinear forms, each being symmetric or alternating, making use of the classification of pairs of linear mappings between vector spaces given by J. Dieudonné.