"Two sided ideal in M2×2(C)" was answered by our community

College
Algebra
Abstract algebra

Ramiro Grant

Answered question

2022-04-09

Two sided ideal in

M_{2 \times 2} (C)

Answer & Explanation

cutimnm135imsa

Beginner2022-04-10Added 21 answers

Step 1
Not even a subring.
$U = (\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}) (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}) (\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}) = (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}) \in B$
$U + I A I = U + A = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) \notin B$
The identity matrix can't be in B because B only contains matrices of rank no more than 1.

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