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Let \(R\) be an associative unital ring with the unit group \(U(R)\). The additive group of \(R\) is denoted by \(R^+\). In [ Rings with finitely many orbits under the regular action, Lecture Notes in Pure and Applied Mathematics 236, Dekker, New York 2004, 343-347], Yasuyuki Hirano concentrated on the left regular group action of \(U(R)\) on \(R^+\) defined by \(a\rightharpoonup x=ax\) for all \(a\in U(R)\), \(x\in R\). The main result of this paper asserted the equivalence of the following statements: (i) \(R\) has only a finite number of orbits under the left regular action of \(U(R)\) on \(R^+\); (ii) \(R\) has only a finite number of left ideals. In the talk we will consider the more general group action of \(U(R)\times U(R)\) on \(R^+\) defined by
\begin{equation}
\label{Eq:1}
(a,b)\rightharpoonup x=axb^{-1},
\end{equation}
for all \(a,b\in U(R)\), \(x\in R\). The action (\ref{Eq:1}) induces in a natural way an action of the group \(U(R)\times U(R)\) on the set of left (respectively, principal left) ideals of \(R\) and of ideals of \(R\), however the action on the latter set is trivial. Orbits under the action (\ref{Eq:1}) are called simply \(U\)-orbits. We introduce the following properties:
FNE - \(R\) has only a finite number of \(U\)-orbits of elements; FNPLI - \(R\) has only a finite number of \(U\)-orbits of principal left ideals; FNLI - \(R\)
has only a finite number of \(U\)-orbits of left ideals; FNI - \(R\) has only a finite number of \(U\)-orbits of ideals (\(R\) has only a finite number of ideals).
We can directly verify the following connections between the above properties:
\begin{equation}
\label{Eq:2}
FNE\Rightarrow FNPLI\Rightarrow FNI\quad{\rm and}\quad FNLI\Rightarrow FNPLI\Rightarrow FNI.
\end{equation}
Since for any division ring \(D\) and any positive integer \(n\), the \(n\times n\) matrix ring \(M_n(D)\) has exactly \(n+1\) \(U\)-orbits both of elements and of left ideals, it follows that
every semisimple artinian ring satisfies all the properties listed in Formula (\ref{Eq:2}). In the talk we will discuss two questions. One of them is:
Under which conditions does a left and/or right artinian ring satisfy FNE or a similar property? The other is:
Must every ring satisfying FNE or a similar property be left and/or right artinian, or at least semiprimary? Continuuing our discussion we will concentrate on
group rings with finite number of orbits.
This talk is an outgrowth of our joint paper with Jan Krempa [On rings with finite number of orbits, Publicacions Matemàtiques 58 (2014), 233--249].
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