A Characterization of Two-Sided Artinian Rings

When R is a two-sided artinian ring, it is shown that if each non-invertible element of R can be expressed as a product of idempotents in R, then R satisfies one of the following:
(1) R≌M_n(D)(n≥1,D is some division ring)
(2) R≌Z₂⊕Z₂⊕...⊕Z₂(s copies, some s≥1).
(3) R/J(R)≌Z₂⊕Z₂, and R is an isostructrure of the upper triangular matrix ring T₂2 over Z₂.