Let $R$ be a commutative $k-$algebra over a field $k$. Assume $R$ is a noetherian, infinite, integral domain. The group of $k-$automorphisms of $R$,i.e.$Aut_k(R)$ acts in a natural way on $(R-k)$.In the first part of this article, we study the struct
ure of $R$ when the orbit space $(R-k)/Aut_k(R)$ is finite.We note that most of the results, not particularly relevent to fields, in [1,S 2] hold in this case as well. Moreover, we prove that $R$ is a field. In the second part, we study a special case of the Conjecture 2.1 in [1] : If $K/k$ is a non trivial field extension where $k$ is algebraically closed and $mid (K-k)/Aut_k(K) mid = 1$ then $K$ is algebraically closed. In the end, we give an elementary proof of [1,Theorem 1.1] in case $K$ is finitely generated over its prime subfield.
Let $R$ be a commutative ring with identity. We define a graph $Gamma_{aut}(R)$ on $ R$, with vertices elements of $R$, such that any two distinct vertices $x, y$ are adjacent if and only if there exists $sigma in aut$ such that $sigma(x)=y$. The ide
a is to apply graph theory to study orbit spaces of rings under automorphisms. In this article, we define the notion of a ring of type $n$ for $ngeq 0$ and characterize all rings of type zero. We also characterize local rings $(R,M) $ in which either the subset of units ($ eq 1 $) is connected or the subset $M- {0}$ is connected in $Gamma_{aut}(R)$.
We give a new proof of the classical result due to Rodney Y. Sharp and Peter Vamos on the dimension of tensor product of a finite number of field extensions of a given field.
We define the notion of a power stable ideal in a polynomial ring $ R[X]$ over an integral domain $ R $. It is proved that a maximal ideal $chi$ $ M $ in $ R[X]$ is power stable if and only if $ P^t $ is $ P$- primary for all $ tgeq 1 $ for the prime
ideal $ P = M cap R $. Using this we prove that for a Hilbert domain $R$ any radical ideal in $R[X]$ which is a finite intersection G-ideals is power stable. Further, we prove that if $ R $ is a Noetherian integral domain of dimension 1 then any radical ideal in $ R[X] $ is power stable. Finally, it is proved that if every ideal in $ R[X]$ is power stable then $ R $ is a field.
