An element $a$ in a ring $A$ with identity is said to be simply polar if there is $b$ for which $a=aba$, with $ab=ba$. If in addition $b=bab$ then such an element $b$ is unique. The question is why $b$ commutes with everything that commutes with $a$.
2026-03-30 02:04:03.1774836243
simply polar elements in a ring
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Assume $ac=ca$. Then $abca=abac=ac=ca=caba=acba$, hence $abc=bac=bca = babca=bacba=abcab = acbab=acb$, i.e. $bc-cb$ is left annihilated by $a$ (and by symmetry is also right annihilated). Then $a(bc-cb+b)a = aba=a$ and $a(bc-cb+b) = ab=ba = (bc-cb+b)a$ and the uniqueness of $b$ implies $bc-cb+b=b$ and finally $bc=cb$.