IN ref. Jacobson proved the structure theorem for primitive rings with nonzero socles that R is a primitive ring with socle S≠{0 } if and only if there is a pair of dual vector spaces ( M, M') over a division ring Δ such that S = F (M, M') is contained in R is contained in D(M, M'), where D(M, M') = { ω G Ω | ωM' is contained in M', Ω is the complete ring of linear transformations of M over Δ }, F(M, M') is the set of all linear transformations of D( M, M') of finite rank. After that, some people reproved this theorem by using different methods such as those in refs. In this note, the author introduces the concept of quasi-element for a subring of the ring of all linear transformations of a vector space, and derives the quasi-critical ring of a primitive ring with nonzero socle. Furthermore, the structure theorem mentioned above is improved.
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