Furthermore, the assumption that scalars can be passed in and out of the bilinear product translates to
Perhaps the most important outcome of these two axioms of an algebra is the opportunity to express polynomial like equations over the algebra. Without the distributive axiom we cannot establish connections between addition and multiplication. Without scalar multiplication we cannot describe coefficients. With these equations we can define certain subalgebras, for example we see both axioms at work in
Given an algebra , the set
is a submodule of .
For now let elements of be denoted with to distinguish them from scalars. As a module for all . Then
Also given then for all ,
Finally, given we have
Although this set appears like a reasonable object to define as the center of an algebra, it is usually preferable to produce a subalgebra, not simply a submodule, and for this we need elements that can be regrouped in products associatively, that is, that lie in the nucleus. So the center is commonly defined as
When the algebra has an identity (unity) then we can go further to identify as a subalgebra of by . Then we see this subalgebra is necessarily in the center of . As a converse, given a unital ring (associativity is necessary), the center of the ring forms a commutative unital subring over which is an algebra. In this way unital rings and associative unital algebras are often interchanged.
|Date of creation||2013-03-22 16:27:20|
|Last modified on||2013-03-22 16:27:20|
|Last modified by||Algeboy (12884)|