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How would one go about checking if a given 2 x 2 matrix is an ideal. I am unclear as to what an ideal is and would like to know the steps in order to make the verification. Also, if it helps, I had proved that S subring of another ring which is all the set of 2 x 2 (integer) matrices while the Matrices in question are

$$ S = \left(\begin{array}{cc} a & 0 \\ b & c \\ \end{array} \right) $$

$$T = \left(\begin{array}{cc} 2a & 2b \\ 2c & 2d \\ \end{array} \right) $$

I did not check if T was a subring.

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You seem to have misunderstood the question, as well as the notion of an ideal.

If $S$ is a subset of a ring $R$, we say that $S$ is an ideal (specifically a two-sided ideal) if

  • $S$ is closed under addition, i.e. for any $s,s' \in S$ we have $s+s' \in S$
  • $S$ is absorbing under multiplication, i.e., for any $s \in S$ and $r \in R$, both $rs$ and $sr$ are elements of $S$.

The set $S$ described is $$ S = \left\{ \pmatrix{a&0\\b&c}:a,b,c \in \Bbb Z \right\} $$ Note that we are given a set of matrices, not "a $2 \times 2$ matrix" as you had put it.

This set of matrices fails to be an ideal. It is a subring, since it is closed under addition and multiplication (and contains the multiplicative identity), but it is not an ideal because it fails to be absorbing under multiplication. In particular, we can find a matrix $s \in S$ and an integer matrix $r$ such that $rs$ is not an element of $S$.

The set $T$ described is $$ T = \left\{ \pmatrix{2a&2b\\2c&2d}:a,b,c,d \in \Bbb Z \right\} $$ This set is an ideal. In order to prove that is an ideal, you need to show that both of the conditions described hold. Whether or not $T$ is a subring depends on your definition.

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An ideal of a ring $R$ is a nonempty subset $I$ such that $I$ is closed under addition, and for all $r \in R$ and $s \in I$, we have $rs \in I$.

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