For basic definitions and examples see here. We continue our discussion with a few more examples.
Example 1. Let be the algebra of matrices with entries in an algebra Let be the matrix with in the -th row and -th column and anywhere else. Let for all integers For we define So, for example, if then and and for all The reason that holds is that for every we may let Then obviously and Besides, every appears in for the unique Finally and is non-zero if and only if Thus
Example 2. Suppose are two groups and is an onto group homomorphism. If is -graded, then is also -graded if we define for all Clearly
It is also easy to see that for all
Example 3. This example is an application of Example 2. Let and Let be the natural group homomorphism. Let with the grading introduced in Example 1. So, by Example 2, we have a -grading for where
and, similarly, is the direct sum of all such that is odd.
We now move on to define an important concept, i.e. a graded ideal of a graded algebra.
Definition. Let be a -graded algebra. A graded or homogenous ideal of is an ideal such that Graded left or right ideals and graded subalgebras of are defined analogously.
Theorem. Let be a -graded algebra and let be an ideal of Then is graded if and only if, as an ideal, is generated by a subset of In other words, is graded if and only if can be generated by a set of homogeneous elements of
Proof. If is generated by then Now, if then for some Let Then where Clearly because and is an ideal of Also, because and Thus and hence So we have proved that and, since the other side of the inclusion is trivial, we get that is graded. Conversely, suppose that and let We want to prove that Obviously because and is an ideal. Also every element of is a finite sum of elements of because Thus and we are done.