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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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 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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