complete ring of quotients
(here is the set of -module morphisms from to ) and define .
Now we shall assign a ring structure to by defining its addition and multiplication. Given two dense ideals and two elements for , one can easily check that and are nontrivial (i.e. they aren’t ) and in fact also dense ideals so we define
It is easy to check that is in fact a commutative ring with unity. The elements of are called .
There is also an equivalence relation that one can define on . Given for , we write
, where is the total quotient ring. One can also in general define complete ring of quotients on noncommutative rings.
- Huckaba J.A. Huckaba, ”Commutative rings with zero divisors”, Marcel Dekker 1988
|Title||complete ring of quotients|
|Date of creation||2013-03-22 16:20:29|
|Last modified on||2013-03-22 16:20:29|
|Last modified by||jocaps (12118)|
|Defines||fraction of rings|
|Defines||complete ring of quotients|