complete ring of quotients
Consider a commutative^{} unitary ring $R$ and set
$$\mathcal{S}:=\{{\mathrm{Hom}}_{R}(I,R):I\text{is dense in}R\}$$ |
(here ${\mathrm{Hom}}_{R}(I,R)$ is the set of $R$-module morphisms^{} from $I$ to $R$) and define $A:={\bigcup}_{B\in \mathcal{S}}B$.
Now we shall assign a ring structure^{} to $A$ by defining its addition and multiplication. Given two dense ideals ${I}_{1},{I}_{2}\subset R$ and two elements ${f}_{i}\in {\mathrm{Hom}}_{R}({I}_{i},R)$ for $i\in \{1,2\}$, one can easily check that ${I}_{1}\cap {I}_{2}$ and ${f}_{2}^{-1}({I}_{1})$ are nontrivial (i.e. they aren’t $\{0\}$) and in fact also dense ideals so we define
${f}_{1}+{f}_{2}\in {\mathrm{Hom}}_{R}({I}_{1}\cap {I}_{2},R)$ by $({f}_{1}+{f}_{2})(x)={f}_{1}(x)+{f}_{2}(x)$
${f}_{1}*{f}_{2}\in {\mathrm{Hom}}_{R}({f}_{2}^{-1}({I}_{1}),R)$ by $({f}_{1}*{f}_{2})(x)={f}_{1}({f}_{2}(x))$
It is easy to check that $A$ is in fact a commutative ring with unity. The elements of $A$ are called .
There is also an equivalence relation^{} that one can define on $A$. Given ${f}_{i}\in {\mathrm{Hom}}_{R}({I}_{i},R)$ for $i\in \{1,2\}$, we write
$${f}_{1}\sim {f}_{2}\iff {f}_{1}|{I}_{1}\cap {I}_{2}={f}_{2}|{I}_{1}\cap {I}_{2}$$ |
(i.e. ${f}_{1}$ and ${f}_{2}$ belong to the same equivalence class^{} iff they agree on the intersection^{} of the dense ideal where they are defined).
The factor ring $Q(R):=A/\sim $ is then called the complete ring of quotients.
Remark.
$R\subset T(R)\subset Q(R)$, where $T\mathit{}\mathrm{(}R\mathrm{)}$ is the total quotient ring. One can also in general define complete ring of quotients on noncommutative rings.
References
- Huckaba J.A. Huckaba, ”Commutative rings with zero divisors^{}”, Marcel Dekker 1988
Title | complete ring of quotients |
---|---|
Canonical name | CompleteRingOfQuotients |
Date of creation | 2013-03-22 16:20:29 |
Last modified on | 2013-03-22 16:20:29 |
Owner | jocaps (12118) |
Last modified by | jocaps (12118) |
Numerical id | 17 |
Author | jocaps (12118) |
Entry type | Definition |
Classification | msc 13B30 |
Related topic | CompleteRingOfQuotientsOfReducedCommutativeRings |
Related topic | EpimorphicHull |
Defines | fraction of rings |
Defines | complete ring of quotients |