Group ring and left zero divisor.

Let $K$ be a finite field and $G$ be a discrete group.

Is it true that for every $a,b\in K[G]$ the condition $ab=0$ implies $ba=0$?

It does not seem to be related to zero divisor problem, any ideas if this can be true and for which fields?

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Can you be more specific for what you mean by $K[G]$? Is this the group ring, or the ring of functions on $G$? – Christopher Drupieski Mar 12 '11 at 15:10
$K[G]$ is the group ring. – Kate Juschenko Mar 12 '11 at 15:26
I don't believe this even for finite $G$. After all a group ring of a finite group $G$ over a field of characteristic coprime to $\left|G\right|$ is semisimple and therefore a product of matrix rings (after corresponding field extension!), which generally are not 1x1 matrix rings. It is easy to find two 2x2 matrices with product $0$ in one order but not $0$ in the other order. – darij grinberg Mar 12 '11 at 15:52
unfortunately, this does not give an example where the property is not satisfied. – Kate Juschenko Mar 12 '11 at 16:05
Dear Kate, here is an explicit example of what Darij says: in ${\mathbb F_5}[S_3]$, $A=1+(23)$ and $B=(12)-(123)$ satisfy $AB=0$ and $BA\ne 0$. – Tim Dokchitser Mar 12 '11 at 16:48

3 Answers

Let G be non-abelian of order 6, with x of order 2 and y of order 3. In such a group yxy = x, since both x and xy have order 2. Let K be a field with 2 elements. Then (x+y)⋅(1+xy) = x+y + y+yxy = x+y + y+x = 0, but (1+xy)⋅(x+y) = x+y + xyx+xyy = x+y + yy + xyy ≠ 0.

You may be thinking of the property: if a⋅b = 0 then there is some non-zero c such that c⋅a = 0. This holds in all (two-sided) Artinian rings (because elements are either units or zero-divisors). I believe this is true for two-sided self-injective rings as well. I don't know if it is possessed by group rings of infinite groups over finite fields.

(Thanks to Greg Marks:) The classification of finite group rings over fields where ab=0 implies ba=0 is given in:

Gutan, Marin; Kisielewicz, Andrzej. "Reversible group rings." J. Algebra 279 (2004), no. 1, 280–291. MR2078399 DOI:j.jalgebra.2004.02.011.

In particular, K is a field of order 22n-1 and G is the quaternion group of order 8, or G is abelian. Li and Parmenter (2007) extend this to finite group rings over commutative rings with 1 in MR2372321.

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also there are some answers in comments – Kate Juschenko Mar 12 '11 at 17:14
G=Q8 over K=GF(2) has the ab=0 implies ba=0 property, but G=D8 over K=GF(2) does not. It seems the characterization might be a little strange. – Jack Schmidt Mar 12 '11 at 17:21
For finite groups $G$, I can at least show that $[G,[G,G]]=1$ if $G$ satisfies the property in question. (Characteristic can be arbitrary.) – darij grinberg Mar 12 '11 at 17:24
The field definitely matters in general, as G=Q8 over K=GF(3) does not have the property. I would be curious to hear a counter-example to: if KG has the ab=0 implies ba=0 property then either G is abelian, G is infinite, or (K has characteristic 2 and G=Q8). – Jack Schmidt Mar 12 '11 at 18:13
You are very kind to credit me, but I doubt I deserve it. I am embarrassed to say that the very interesting paper of Gutan and Kisielewicz had slipped my mind. Gutan and Kisielewicz completely resolve the group algebra reversibility question for the case of torsion groups. They remark that the torsion-free case is connected with the zero divisor problem, in the sense that it will be very difficult to produce a non-reversible group algebra with a torsion-free group, since this will of course resolve the zero divisor problem in the negative. – Greg Marks Mar 14 '11 at 2:08

The condition $ab=0 \Rightarrow ba=0$ defines what are often called reversible rings, which, for example, have the property that the set of nilpotent elements is an ideal that coincides with the prime radical.  A full matrix ring can't have this property, so you can construct counterexamples by taking any finite field $K$ and nonabelian group $G$ to which Maschke's theorem applies.  An alternative example of a non-reversible group algebra is $K[G]$ where $K$ is the field of two elements and $G$ is the dihedral group of order $8$.  Here the set of nilpotent elements does coincide with the prime radical (the ring is local artinian), but one can find elements $a,b \in K[G]$ with $ab=0$ but $ba \neq 0$.

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Using the miniscule amounts of representation theory I know, you can construct an example as follows.

Let $G=S_3=D_6=\left< r,s\mid r^3=s^2=srsr=e \right>$ (rotation $r$ and flip $s$), and let $K=\mathbb F_5[\omega]$ where $\omega^3=1$ (finite field!). Then an irreducible representation of $G$ is given by the $2$-dimensional vector space $V$ with basis $\{v, sv\}$ and action of $G$ on $V$ $\rho\colon G\to Aut(V)$ generated by by $\rho(s)(v)=sv$, $\rho(s)(sv)=v$, $\rho(r)v=\omega v$ and $\rho(r)(sv)=\omega^2 sv$.

Hence, we have $\rho(e)=\left[\begin{matrix}1&0\newline 0&1\end{matrix}\right]$, $\rho(r)=\left[\begin{matrix}\omega&0\newline 0&\omega^2\end{matrix}\right]$, $\rho(r^2)=\left[\begin{matrix}\omega^2&0\newline 0&\omega\end{matrix}\right]$, $\rho(s)=\left[\begin{matrix}0&1\newline 1&0\end{matrix}\right]$, $\rho(sr)=\left[\begin{matrix}0&\omega^2\newline \omega&0\end{matrix}\right]$, $\rho(sr)=\left[\begin{matrix}0&\omega\newline \omega^2&0\end{matrix}\right]$.

The first four matrices are linearly independent over $\mathbb K$ and hence generate $End_{\mathbb K}(V)$ which is $4$-dimensional. Now we know that if $\bar K$ is the algebraic closure of $K$, then $\bar K[G]=\bigoplus_W End_{\bar{\mathbb K}}(W)$ where $W$ are irreducible representations over $\bar K$. Character theory tells us that the projection of $\bar K[G]$ onto $End_{\bar {\mathbb K}}(W)$ is given by left-multiplication by the idempotent $\frac{\dim W}{|G|}\sum_{g\in G}\chi_W(g^{-1})g$ where $\chi_W$ is the trace of $\rho_W\colon G\to Aut(W)$.

So in particular we have the idempotent $\phi=\frac13(2e-r-r^2)\in D_6$, which projects $\bar K[G]$ onto $End_{\bar{\mathbb K}}(V)$, and hence must project $K[G]$ into $End_{\mathbb K}(V)$. But we see that $\phi(e)=\frac13(2e-r-r^2)$, $\phi(r)=\frac13(-e+2r-r^2)$, $\phi(r^2)=\frac13(-e-r+2r^2)$, and $\phi(s)=\frac13(2s-rs-r^2s)$ are four linearly independent elements in the image of $\psi$ and thus span $End_{\mathbb K}(V)$.

So take your favorite $2\times 2$ matrices $A$ and $B$ (over $K)$ such that $AB=0$ but $BA\neq 0$, find $a_g$ and $b_g$ such that $A=\sum_{g\in \{e,r,r^2,s\}} a_g\rho(g)$ and $B=\sum_{g\in \{e,r,r^2,s\}} b_g\rho(g)$, and then $a=\sum_{g\in \{e,r,r^2,s\}}a_g\phi(g)$ and $b=\sum_{g\in \{e,r,r^2,s\}}b_g\phi(g)$ will be such that $ab=0$ but $ba\neq 0$.

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