- Rings of fractions and Goldie's theorems

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# Rings of fractions and Goldie's theorems

Author

Yew, Angela Choy Leng

Supervisor

Teo, Kok Ming

Abstract

n commutative ring theory, we can always construct the field of quotients of an integral domain. Then we generalize to the ring of fractions formed using a commutative ring R and a multiplicative set X of R. The elements of this ring of fractions RX-1 has the form r/x where r Є R and x Є R and r/x=s/y if and only if (ry-sx)z = 0 for some z Є X.

In the case of noncommutative rings, the construction is not as simple. The general question is 'Given a noncommutative ring R and a multiplicative subset X of R, can we always form the ring of fractions of R such that every element has the form rx-1 for some r, x Є R, x ≠ 0?'. In order to answer this question, we need to look for conditions on X such that, the elements of X can be inverted, as well as having our end results to resemble the rings of fractions constructed in the commutative case.

In conclusion, it turns out that X must be a right denominator set in order for the right ring of fractions to exist. For the case where R is a semiprime right Goldie ring and X is the set of regular elements of R, then the right ring of fractions constructed is a semi-simple ring. This is known as the Goldie's Theorem.

In the case of noncommutative rings, the construction is not as simple. The general question is 'Given a noncommutative ring R and a multiplicative subset X of R, can we always form the ring of fractions of R such that every element has the form rx-1 for some r, x Є R, x ≠ 0?'. In order to answer this question, we need to look for conditions on X such that, the elements of X can be inverted, as well as having our end results to resemble the rings of fractions constructed in the commutative case.

In conclusion, it turns out that X must be a right denominator set in order for the right ring of fractions to exist. For the case where R is a semiprime right Goldie ring and X is the set of regular elements of R, then the right ring of fractions constructed is a semi-simple ring. This is known as the Goldie's Theorem.

Date Issued

2004

Call Number

QA247 Yew

Date Submitted

2004