Answer to Question #17114 in Abstract Algebra for Tsit Lam
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”?
The statement is indeed true. Ahomomorphic image S of A = Mm(R) has the form A/I where I is an ideal of A. But I = Mm(A) for some ideal A ⊆ R. Therefore, S ∼ Mm(R)/Mm(A)∼ Mm(R/A).If m ≥ 2, then S is indeed a matrix ring.
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