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Abstract Algebra

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Commutative Rings: Definition and Examples

ABSALG-ZIKNQ2

A ring $R$ is commutative if for each $x, y \in R$, we have $xy = yx$.

Which of the following rings are commutative?

Select ALL that apply.

A

$\mathbb{H}$ (the quaternions)

B

$\mathbb{Z}$

C

$\mathbb{Z}/n\mathbb{Z}$

D

The matrix ring $M_2(\mathbb{Z})$

E

The $2 \times 2$ diagonal matrices over $\mathbb{Z}$