Let

be a ring and

an element of

. Let

.

is a subring of

.

This may depend on the definition. Many authors require that if R is unitary then in order to be considered a candidate to be a subring a subset must contain the unit of R. In this case, though, it must be not so since then a = 0.
Let

(the ring of 2×2 matrices with entries from the field

) and

. Determine the elements of the subring S defined previously.

**(I)** How many elements are in S?

**(II)** Show that S is not an ideal of R.

__Attempt:__ **(I)** the elements of S are 2x2 matrices x such that

.

let

If I solve the homogeneous system

4a+2c=0

4b+2d=0

3a+4c=0

3b+4d=0

I get a=b=c=d=0.

Even without checking this cannot be right since the matrix x is singular, and indeed: any matrix of the form belongs to
So does this mean the only element in the subring S is the zero matrix?

**(II)** I will use the "ideal test":

Since the zero matrix belongs to S,

.

Then

and

. (x,y are in S).

The question says show that S is

**NOT** an ideal. But the problem is that S passes the ideal test since (x-y) is equal to the zeo matrix and therefore

. The same is true with multipication.

No, it's not. Try to find a counterexample to now that you know the general form of elements in Tonio
So, is something wrong with the question or did I make a mistake?!