A characterization of the Jacobson radical in ternary algebras

Abstract:

The Jacobson radical Rad $T$ for a ternary algebra $T$ is characterized as one of the following: (i) the set of properly quasi-invertible elements in $T$; (ii) the set of $x \in T$ such that the principal right ideal $\left \langle {xTT} \right \rangle$ or left ideal $\left \langle {TTx} \right \rangle$ is quasi-regular in $T$; (iii) the unique maximal quasi-regular ideal in $T$; (iv) the set of $x \in T$ such that Rad ${T^{(x)}} = {T^{(x)}}$. We also obtain ternary algebra-analogs of characterization of the radicals of certain subalgebras in an associative algebra.