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- Jun 22, 2012

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I am reading Watson: Topics in Commutative Ring Theory.

in Ch 3: Localization, Watson defines the quotient field of an integral domain as follows:

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We begin by defining an equivalence relation on an integral domain D. Let D be an integral domain. We define an equivalence relation on the set S of "fractions" using elements of D,

S = {a/b | a, b \(\displaystyle \in \) D , \(\displaystyle b \ne 0 \)} ,

by a/b \(\displaystyle \cong \) c/d if and only if ad = bc

... ...

we turn the set of equivalence classes into a ring (actually a field) by defining addition and multiplication as follws:

[a/b] + [c/d] = [(ad + bc)/bd]

and

[a/b] . [c/d] = [ac/bd].

In this way, Watson has explained his Definition 6.3 which reads as follows:

Watson, then generalises the above process from an integral domain to a ring R by restricting the denominators to regular elements (ie elements that are not zero divisors)). This results in the formation of the total quotient ring defined as follows:

Watson then generalises the process further in a process called "localization" where now denominators are allowed to be any element from a multiplicative system (or multiplicatively closed set) defined as follws:

Watson then writes:

If T is a multiplicative system of a ring R, then an equivalence relation can be defined on an appropriate set S of "fractions" using elements of R

S = {a/b | a,b \(\displaystyle \in \) R, and b \(\displaystyle \in \) T}

and

a/b \(\displaystyle \cong \) c/d if and only if r(ad - bc) =0

for some \(\displaystyle t \in T \) ... ...

Watson argues that this also results in a ring ... but my problem with this construction of a ring of fractions is that the original ring R is a multiplicative system - so then one possibility is that T = R - but then how do we avoid the problem of zero divisors -i.e. in addition and multiplication we may end up with bd = 0 for b and d not equal to zero.

Can someone please clarify this issue for me?

Peter

in Ch 3: Localization, Watson defines the quotient field of an integral domain as follows:

--------------------------------------------------------------------------------------------------

We begin by defining an equivalence relation on an integral domain D. Let D be an integral domain. We define an equivalence relation on the set S of "fractions" using elements of D,

S = {a/b | a, b \(\displaystyle \in \) D , \(\displaystyle b \ne 0 \)} ,

by a/b \(\displaystyle \cong \) c/d if and only if ad = bc

... ...

we turn the set of equivalence classes into a ring (actually a field) by defining addition and multiplication as follws:

[a/b] + [c/d] = [(ad + bc)/bd]

and

[a/b] . [c/d] = [ac/bd].

**(***Note that the right hand sides of these expressions make sense because D is a domain and so**__\(\displaystyle bd \ne 0 \)__***)In this way, Watson has explained his Definition 6.3 which reads as follows:

**Definition 6.3**Let D be an integral domain. The above field F of equivalence classes of fractions from D, with addition and multiplication defined as above, is called the**quotient field**of D.Watson, then generalises the above process from an integral domain to a ring R by restricting the denominators to regular elements (ie elements that are not zero divisors)). This results in the formation of the total quotient ring defined as follows:

**Definition 6.5**Let R be a ring. The ring Q(R) of of equivalent classes of fractions from R whose denominators are regular elements, with addition and multiplication defined as above is called a**total quotient ring**of R.Watson then generalises the process further in a process called "localization" where now denominators are allowed to be any element from a multiplicative system (or multiplicatively closed set) defined as follws:

**Definition 6.6**Let R be a ring. A subset T of R is a**multiplicative system**if \(\displaystyle 1 \in T \) and if \(\displaystyle a, b \in T \) implies that \(\displaystyle ab \in T \) - that is T is multiplicatively closed and contains 1.Watson then writes:

If T is a multiplicative system of a ring R, then an equivalence relation can be defined on an appropriate set S of "fractions" using elements of R

S = {a/b | a,b \(\displaystyle \in \) R, and b \(\displaystyle \in \) T}

and

a/b \(\displaystyle \cong \) c/d if and only if r(ad - bc) =0

for some \(\displaystyle t \in T \) ... ...

Watson argues that this also results in a ring ... but my problem with this construction of a ring of fractions is that the original ring R is a multiplicative system - so then one possibility is that T = R - but then how do we avoid the problem of zero divisors -i.e. in addition and multiplication we may end up with bd = 0 for b and d not equal to zero.

Can someone please clarify this issue for me?

Peter

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