# Answer to Question #17272 in Algebra for sanches

Question #17272

Show that rad R is the intersection A of the annihilators of all simple left R-modules.

Expert's answer

We show rad

*R*=*A*. For any modular maximal left ideal m,we have (rad*R*)*R**⊆*rad*R**⊆*m. Therefore, (rad*R*)*· R/*m = 0, sorad*R**⊆ A*. Conversely, if*a**∈ A*, then*aR**⊆*m for any modular maximal left ideal, so*aR**⊆*rad*R*. Since rad*R*is quasi-regular,so is*aR*; hence*a**∈*rad*R*.Need a fast expert's response?

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