explicit definition of polynomial rings in arbitrarly many variables
Let be a ring and let be any set (possibly empty). We wish to give an explicit and formal definition of the polynomial ring .
We start with the set
If then the elements of can be interpreted as monomials
Now we will define multiplication. First note that we have a multiplication on . For any put
This is the same as multiplying .
Now for any define
Now if then we define the multiplication
Note that all of this well-defined, since both and vanish almost everywhere.
which is defined as follows: for any let be the function such that if is such that for any , then put and for any other function put . Then
is the identity in . Anyway we can always interpret as a subset of if put for .
Also note, that itself can be interpreted as a subset of . Indeed, for any define
by and for any . Then define
by putting and for any . It can be easily seen that if and only if . Thus we will use convention .
|Title||explicit definition of polynomial rings in arbitrarly many variables|
|Date of creation||2013-03-22 19:18:10|
|Last modified on||2013-03-22 19:18:10|
|Last modified by||joking (16130)|