Question

Let R = $M_2(\mathbb{Z})$ be the ring of a 2 X 2 matrices with integer entries. For any $n \in \mathbb{Z}$, prove that the set $S = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, \text{and } d \text{ all are divisible by } n \right\}$ is a two sided Ideal.

          Let R = $M_2(\mathbb{Z})$ be the ring of a 2 X 2 matrices with integer entries.
For any $n \in \mathbb{Z}$, prove that the set
$S = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, \text{and } d \text{ all are divisible by } n \right\}$ is a two sided Ideal.

Let R = M2(ℤ) be the ring of a 2 X 2 matrices with integer entries.
For any n ∈ℤ, prove that the set
S = {
< p m a t r i x >
: a, b, c, and d all are divisible by n } is a two sided Ideal.

Added by Julie C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/14/2023

Step 1

To prove that S is a subset of R, we need to show that every element in S is also an element of R. In this case, every element in S is a 2x2 matrix with integer entries. Since R is defined as the ring of 2x2 matrices with integer entries, every element in S is Show more…

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Let R = M,(z) be the ring of a 2 X 2 matrices with integer entries For any n e z , prove that the set S=1 ) : a,b, c, and d all are divisible by n} is a two sided Ideal.