0.1 Matrix Rings
A ring is said to be a matrix ring if there is a ring and a positive integer such that
the ring of matrices with entries as elements of . Usually, we simply identify with .
Generally, one is interested to find out if a given ring is a matrix ring. By setting , we see that every ring is trivially a matrix ring. Therefore, to exclude these trivial cases, we call a ring a trivial matrix ring if there does not exist an such that . Now the question becomes: is a non-trivial matrix ring?
Actually, the requirement that be a ring in the above definition is redundent. It is enough to define to be simply a set with two binary operations and . Fix a positive integer , define the set of formal matrices with coefficients in . Addition and multiplication on are defined as the usual matrix addition and multiplication, induced by and of respectively. By abuse of notation, we use and to denote addition and multiplication on . We have the following:
The first two assertions above are easily observed. To see how the last one roughly works, assume is the multiplicative identity of . Next define to be the matrix whose -cell is and everywhere else. Using cell entries from , we solve the system of equations
to conclude that takes the form of a diagonal matrix whose diagonal entries are all the same element . Furthermore, this is an idempotent. From this, it is easy to derive that is in fact a multiplicative identity of (multiply an element of the form , where is an arbitrary element in ). The converse of all three assertions are clearly true too.
Any ring is Morita equivalent to the matrix ring for any positive integer .
0.2 Matrix Groups
Suppose is unital. , the group of units of , being isomorphic to the group of automorphisms of , is called the general linear group of . A matrix group is a subgroup of for some matrix ring . If is a field, in particular, the field of real numbers or complex numbers, matrix groups are sometimes also called classical groups, as they were studied as far back as the 1800’s under the name groups of tranformations, before the formal concept of a group was introduced.
|Date of creation||2013-03-22 15:54:18|
|Last modified on||2013-03-22 15:54:18|
|Last modified by||CWoo (3771)|