Question

Which of the following statements is/are false?
(I) \(I=\left\{\begin{pmatrix}a&0\\c&d\end{pmatrix}: a,c,d\in\mathbb{R}\right\}\) is both a subring and an ideal of \(M_2(\mathbb{R})\).
(II) \(\mathbb{Z}[\sqrt2]=\{a+b\sqrt2 : a,b\in\mathbb{Z}\}\) is an integral domain, but not a field.

• A. Only (I)
• B. Only (II)
• C. Both (I) and (II)
• D. Neither (I) nor (II)

Hint:

Test whether \(I\) absorbs multiplication by an arbitrary matrix from \(M_2(\mathbb{R})\); use the norm \(a^2-2b^2\) for \(\mathbb{Z}[\sqrt2]\).

Answer 1

The Correct Option is A

Step 1: Let \(I=\left\{\begin{pmatrix}a&0\\c&d\end{pmatrix}\right\}\), the lower triangular matrices. For two such matrices, \(\begin{pmatrix}a&0\\c&d\end{pmatrix}\begin{pmatrix}a'&0\\c'&d'\end{pmatrix}=\begin{pmatrix}aa'&0\\ca'+dc'&dd'\end{pmatrix}\), which again has zero (1,2)-entry. Together with closure under addition and subtraction, this shows \(I\) is a subring of \(M_2(\mathbb{R})\).

Step 2: Test whether \(I\) is a left ideal. Take \(M=\begin{pmatrix}0&1\\0&0\end{pmatrix}\in M_2(\mathbb{R})\) and \(X=\begin{pmatrix}0&0\\0&1\end{pmatrix}\in I\). Then \(MX=\begin{pmatrix}0&1\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&1\end{pmatrix}=\begin{pmatrix}0&1\\0&0\end{pmatrix}\), which has a nonzero (1,2)-entry, so \(MX\notin I\).

Step 3: Since \(MX\notin I\) for this choice, \(I\) is not a left ideal, hence not an ideal of \(M_2(\mathbb{R})\). So statement (I) is FALSE, since \(I\) is a subring but not an ideal.

Step 4: Now examine \(\mathbb{Z}[\sqrt2]\). It is a subring of \(\mathbb{R}\), and since \(\mathbb{R}\) has no zero-divisors, neither does \(\mathbb{Z}[\sqrt2]\); hence it is an integral domain.

Step 5: Check whether it is a field. The element \(2\in\mathbb{Z}[\sqrt2]\) has inverse \(1/2\) in \(\mathbb{R}\), but \(1/2\) cannot be written as \(a+b\sqrt2\) with \(a,b\in\mathbb{Z}\). So \(2\) has no inverse inside \(\mathbb{Z}[\sqrt2]\), and it is not a field.

Step 6: So statement (II) is TRUE. Only statement (I) is false.

\[\boxed{\text{Only (I) is false}}\]