We describe a correspondence between ${\text{GL}}_n$-invariant tensors and graphs. We then show how this correspondence accommodates various types of symmetries and orientations.

Let $G$ be a group with identity $e$ and let $R$ be a $G$-graded ring. In this paper, we introduce and study the concept of graded $(2,n)$-ideals of $R$. A proper graded ideal $I$ of $R$ is called a graded $(2,n)$-ideal of $R$ if whenever $rst\in I$ where $r,s,t\in h\left(R\right)$, then either $rt\in I$ or $rs\in Gr\left(0\right)$ or $st\in Gr\left(0\right)$. We introduce several results concerning $gr$-$(2,n)$-ideals. For example, we give a characterization of graded $(2,n)$-ideals and their homogeneous components. Also, the relations between graded $(2,n)$-ideals and others that already exist, namely, the graded prime ideals,...

We consider brave new cochain extensions F(BG +,R) → F(EG +,R), where R is either a Lubin-Tate spectrum E n or the related 2-periodic Morava K-theory K n, and G is a finite group. When R is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a G-Galois extension in the sense of John Rognes, but not always faithful. We prove that for E n and K n these extensions are always faithful in the K n local category. However, for a cyclic p-group $C_{p^r}$, the cochain extension $F(B{C}_{p^r+},E_n)\to F(E{C}_{p^r+},E_n)$ is not a Galois...

Let $f:A\to B$ and $g:A\to C$ be two ring homomorphisms and let $J$ and $J^{\text{'}}$ be ideals of $B$ and $C$, respectively, such that $f^{-1}\left(J\right)=g^{-1}\left(J^{\text{'}}\right)$. In this paper, we investigate the transfer of the notions of Gaussian and Prüfer rings to the bi-amalgamation of $A$ with $(B,C)$ along $(J,J^{\text{'}})$ with respect to $(f,g)$ (denoted by $A{\bowtie }^{f,g}(J,J^{\text{'}})),$ introduced and studied by S. Kabbaj, K. Louartiti and M. Tamekkante in 2013. Our results recover well known results on amalgamations in C. A. Finocchiaro (2014) and generate new original examples of rings possessing these properties.

We study the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex. Some results that track the behavior of Gorenstein properties over local ring homomorphisms under composition and decomposition are given. As an application, we characterize a dualizing complex for $R$ in terms of the finiteness of the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex.