Definition 1. A ring with 1 is called simple if and are the only two-sided ideals of
Remark 1. The center of a simple ring is a field.
Proof. Let be any non-zero element of the center of Then is a non-zero two-sided ideal of and hence, since is simple, Thus there exists some such that i.e. is invertible. Since is in the center, is in the center too and we’re done.
Obviously commutative simple rings are just fields. So only non-commutative simple rings are interseting.
Note 1. If is the center of a simple ring then is both a vector space over and a ring. So it is a -algebra. The term simple algebras is commonly used instead of simple rings.
We now give the first example of simple rings.
Example 1. Let be a ring with 1 and let be the ring of matrices with entries from It is a well-known fact, and easy to prove, that is a two-sided ideal of if and only if for some two-sided ideal of In particular, is simple if and only if is simple. So, for example, since every division ring is simple, is simple too. In fact, by the Wedderburn-Artin theorem, a ring is simple and (left or right) Artinian if and only if for some division ring and some integer
Example 1 gives all simple rings which are Artinian. But what about simple rings which are not Artinian? But first some definitions and remarks. From now on we will assume that is a field and is a -algebra.
Definition 2. A -linear map is called a derivation of if for all If there exists some such that for all then is called an inner derivation.
Note that if , then the map defined by , for all , is a derivation.
Remark 2. If is a derivation of then and thus Therefore if then
Remark 3. Consider the polynomial algebra and define the map by Then is a derivation which is not inner. The reason is that if there was such that for all then, since is central, we’d get which is non-sense.
Definition 3. Let be a derivation of An ideal of is called a –ideal if If the only -ideals of are and we’ll say that is –simple.
Obviously if is simple, then is -simple for every derivation
Remark 4. If is an inner derivation of , then every ideal of is also a -ideal. An obvious example of a -simple algebra is with
Defintion 4. If is a derivation of then we define the -algebra to be the set of all polynomials in the indeterminate and left coefficients in , with the usual addition and multiplication and the rule The algebra is also called a differntial polynomial algebra.
Note 2. So a typical element of is in the form where When we multiply two of these polynomials we will have to use the rule given in Definition 4. For example
Of course, we need to prove that is a -algebra but we won’t do it here.
Remark 5. Let and be an inetger. Then an easy induction shows that in we have
where So is a (left) polynomial of degree with the leading coefficient
In part (2) we will give three important examples of simple rings.