.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.
(I) the elements of S are 2x2 matrices x such that .
If I solve the homogeneous system
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
So, is something wrong with the question or did I make a mistake?!