The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in [15]. In the present article, we employ the recently developed F-method, [10], [11] to the couple of non-compatible Lie algebras $\mathrm{Lie}{G}_2\stackrel{i}{\hookrightarrow }so\left(7\right)$, and generalized conformal $so\left(7\right)$-Verma modules of scalar type. As a result, we classify the $i\left(\mathrm{Lie}{G}_2\right)\cap 𝔭$-singular vectors for this class of $so\left(7\right)$-modules.

We present some facts, observations and remarks concerning the problem of finiteness of the rings of constants for derivations of polynomial rings over a commutative ring k containing the field ℚ of rational numbers.

For a finite commutative ring $R$ and a positive integer $k⩾2$, we construct an iteration digraph $G(R,k)$ whose vertex set is $R$ and for which there is a directed edge from $a\in R$ to $b\in R$ if $b=a^k$. Let $R=R_1\oplus ...\oplus R_s$, where $s>1$ and $R_i$ is a finite commutative local ring for $i\in \{1,...,s\}$. Let $N$ be a subset of $\{R_1,\cdots ,R_s\}$ (it is possible that $N$ is the empty set $\varnothing$). We define the fundamental constituents $G_N^{*}(R,k)$ of $G(R,k)$ induced by the vertices which are of the form $\{(a_1,\cdots ,a_s)\in R:a_i\in \mathrm{D}\left(R_i\right)$ if $R_i\in N$, otherwise $a_i\in \mathrm{U}\left(R_i\right),i=1,...,s\},$ where U$\left(R\right)$ denotes the unit group of $R$ and D$\left(R\right)$ denotes the zero-divisor set of $R$. We investigate...

In this paper we compute injective, projective and flat dimensions of inverse polynomial modules as $R\left[x\right]$-modules. We also generalize Hom and Ext functors of inverse polynomial modules to any submonoid but we show Tor functor of inverse polynomial modules can be generalized only for a symmetric submonoid.

Let $𝒩=N_n\left(R\right)$ be the algebra of all $n\times n$ strictly upper triangular matrices over a unital commutative ring $R$. A map $\varphi$ on $𝒩$ is called preserving commutativity in both directions if $xy=yx\iff \varphi \left(x\right)\varphi \left(y\right)=\varphi \left(y\right)\varphi \left(x\right)$. In this paper, we prove that each invertible linear map on $𝒩$ preserving commutativity in both directions is exactly a quasi-automorphism of $𝒩$, and a quasi-automorphism of $𝒩$ can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.