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Abstract Algebra

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Moderate

Matrix Rings and Ideals

ABSALG-IEBDQN

Let $R$ be a ring. The $n\times n$ matrix ring over $R$, denoted by $M_n(R )$, consists of the set of all $n\times n$ matrices with entries from $R$, together with matrix addition and matrix multiplication.

Which of the following sets are either an ideal or a subring of $M_n(\mathbb{Q})$?

A

The set of all diagonal matrices

B

The set of all singular matrices

C

The set of all nonsingular matrices

D

The set of matrices with even integer entries