Nonunital decompositions of the matrix algebra of order three

Abstract

All decompositions of $M_3\left(\mathrm{ℂ}\right)$ into a direct vector-space sum of two subalgebras such that none of the subalgebras contains the identity matrix are classified. Thus, the classification of all decompositions of $M_3\left(\mathrm{ℂ}\right)$ into a direct vector-space sum of two subalgebras as well as description of Rota–Baxter operators of nonzero weight on $M_3\left(\mathrm{ℂ}\right)$ is finished.