Deveno said:if there IS an identity, wouldn't it have to be the same identity as M2(Z) has?
(note: some people require a ring to have identity, some people don't. by "identity" i mean multiplicative identity, as every ring MUST have a 0).
Deveno said:is (R,+) closed under matrix addition? if A is in R, is -A in R? is R closed under matrix multiplication? these are the crucial questions.
A subring is a subset of a larger ring that is itself a ring with the same operations and identity element as the larger ring.
To prove that R is a subring of M2(Z), we must show that R satisfies the three conditions for a subring: closure under addition and multiplication, and containing the identity element. This can be done by showing that any element in R can be added, multiplied, and has an inverse within R, and that the identity element of M2(Z) is also in R.
One example of a subring of M2(Z) is the set of all 2x2 matrices with entries that are multiples of 3. This subset is closed under addition and multiplication, and contains the identity element of M2(Z).
Proving that R is a subring of M2(Z) can be useful in fields such as abstract algebra, number theory, and cryptography. It allows us to study properties of R as a subset of a larger ring, and to use theorems and techniques from the larger ring on R.
If R does not meet the requirements to be a subring of M2(Z), then it is not considered a subring. This means that it may not have the same operations and identity element as the larger ring, and thus cannot be studied using theorems and techniques from the larger ring.