Left, right, and inner socles of associative systems. (Q705728)

Let \(A\) be arbitrary, not necessarily semiprime, associative system. Denote by \(\text{Soc}^1_\ell A\), \(\text{Soc}^1_rA\) and \(\text{Soc}^1_{in}A\) the sums of all minimal left, right and inner ideals. The aim of this paper is to study the relations between these socles. For example, the authors prove that \(\text{Soc}_\ell A\) and \(\text{Soc}_rA\) are contained in \(\text{Soc}_{in}A\). Moreower, if \(A\) is semiprime, then these three socles coincide. If \(A\) is an associative pair or triple system, then \(\text{Soc}_{in}A\) and \(\text{Soc}^1_{in}A\) are ideals of \(A\). Furthermore, if \(A\) is an associative algebra, then all socles are ideals of \(A\).