Basic facts about matrix algebrasPosted on May 25, 2019
Let be a ring, and let be positive integers.
Proposition 1. The -algebras and are isomorphic.
The obvious map between is clearly a bijection and can be seen to be an -algebra homomorphism by observing that multiplication works as expected in each.
Proposition 2. If is an -algebra, then there is a natural isomorphism
Let be defined by This map is clearly an -algebra homomorphism.
Let be the matrix with if and otherwise. Observe that the map is onto as the matrices generate as an -algebra.
Suppose that Then either , , or for all Writing we see that Thus is injective and hence an isomorphism.
Proposition 3. If is an ideal of , let denote the subset of consisting of matrices with entries in The identification is a bijection between the set of two-sided ideals of the ring and the set of two-sided ideals of
First observe that if is a two sided ideal of then as defined above is clearly a two-sided ideal of Now suppose that is a two sided ideal of and define It is easy to see that is a two sided ideal of since is a two sided ideal of as follows. Suppose that and By the definition of there exists a matrix with as an entry. We then easily define a matrix with one nonzero entry equal to such that is an entry of Hence A similar argument shows
It now suffices to show that and for any two sided ideals and It is immediate from definitions that To show the second equality, first observe that by definition. For any , by matrix multiplication we can construct a matrix (with defined as above). Hence we can generate all matrices in we can construct a matrix
Proposition 4. Let denote the identity matrix in The map defined by identifies with the centre of
It is easy to see that if a matrix has nozero entries off of the diagonal, then we may construct a matrix that does not commute with it. If a diagonal matrix has two differring entries along the diagonal, then this matrix does not commute with the matrix consisting of a single nonzero column of ’s in the position matching one of the distinct entries. Thus the only matrices with a chance of commuting are diagonal matrices of the form Thus the map identifies the centre with the centre of
Proposition 5. If is a central simple algebra over a field , then is also a central simple algebra over
By proposition 3 we see that is simple iff is simple. By proposition 4 we see that we may identify with the subring of and that has center equal to the center of Thus if is a CSA over then so is