Let $p$ be a prime number and $n$ a positive integer. I want to know what is the (right) socle of the group ring $A=\mathbb Z_{(p)}C_n$, where $\mathbb Z_{(p)}$ is the localization of integers at the prime ideal $(p)$, and $C_n$ is the cyclic group of order $n$. Thanks for any help!
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
6
It's zero. If $M$ is any non-zero right ideal, then $pM$ is a strictly smaller right ideal, so there are no minimal right ideals.
answered Mar 13, 2017 at 8:53
-
$\begingroup$ Thanks for the answer! Is the same procedure of yours applcable for an arbitrary group ring $R[G]$? $\endgroup$ Commented Mar 13, 2017 at 9:57
-
$\begingroup$ @karparvar It depends on $R$. If $G$ is a finite group, then $RG$ has non-zero socle if and only if $R$ has non-zero socle. $\endgroup$ Commented Mar 13, 2017 at 11:29
-
$\begingroup$ How could one determine the, say, minimal right ideals of $RG$ through those of $R$ to determine the right socle of the former? Thanks, in advance! $\endgroup$ Commented Mar 13, 2017 at 17:51
-
$\begingroup$ @karparvar Depending on exactly what you mean by "determne", that's a hard problem in general. For example, if $R=\mathbb{F}_p$ and $G$ is the symmetric group $S_n$, then it's an unsolved problem to find the dimensions of the minimal right ideals of $RG$. $\endgroup$ Commented Mar 14, 2017 at 8:42
-
1$\begingroup$ @karparvar If $I$ is a minimal ideal of $R$ then $I\sum_{g\in G}g$ is a minimal ideal of $RG$. If $R$ has no minimal ideal, then for any non-zero ideal $J$ of $RG$ there is some non-zero ideal $I$ of $R$ with $0<J\cap IG<J$, so $RG$ has no minimal ideal. $\endgroup$ Commented Mar 14, 2017 at 17:40