Answer to Question #17114 in Abstract Algebra for Tsit Lam

Question

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”?

Answer 1

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_m(A) for some ideal A ⊆ R. Therefore, S ∼ M_m(R)/M_m(A)∼ M_m(R/A).If m ≥ 2, then S is indeed a matrix ring.