# Answer to Question #17114 in Abstract Algebra for Tsit Lam

Question #17114

Let us call a ring A a matrix ring if A ∼ Mm(R) for some integer m ≥ 2and some ring R. True or False: “A homomorphic image of a matrix ring is also a matrix ring”?

Expert's answer

The statement is indeed true. Ahomomorphic image

*S*of*A*= M*(*_{m}*R*) has the form*A/I*where*I*is an ideal of*A*. But*I*= M*(A) for some ideal A*_{m}*⊆**R*. Therefore,*S**∼*M*(*_{m}*R*)*/*M*(A)*_{m}*∼*M*(*_{m}*R/*A)*.*If*m ≥*2, then*S*is indeed a matrix ring.
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