Total quotient ring
In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R that may have zero divisors. The construction embeds R in a larger ring, giving every non-zero-divisor of R an inverse in the larger ring. Nothing more in A can be given an inverse, if one wants the homomorphism from A to the new ring to be injective.
Let be a commutative ring and let be the set of elements which are not zero divisors in ; then is a multiplicatively closed set. Hence we may localize the ring at the set to obtain the total quotient ring .
If is a domain, then and the total quotient ring is the same as the field of fractions. This justifies the notation , which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain.
Since in the construction contains no zero divisors, the natural map is injective, so the total quotient ring is an extension of .
The total quotient ring of a product ring is the product of total quotient rings . In particular, if A and B are integral domains, it is the product of quotient fields.
In an Artinian ring, all elements are units or zero divisors. Hence the set of non-zero divisors is the group of units of the ring, , and so . But since all these elements already have inverses, .
The same thing happens in a commutative von Neumann regular ring R. Suppose a in R is not a zero divisor. Then in a von Neumann regular ring a=axa for some x in R, giving the equation a(xa-1)=0. Since a is not a zero divisor, xa=1, showing a is a unit. Here again, .
- The rational functions over a ring R[dubious ] can be constructed from the polynomial ring R[x] as a total quotient ring.
- In algebraic geometry one considers a sheaf of total quotient rings on a scheme, and this may be used to give one possible definition of a Cartier divisor.
If is a commutative ring and is any multiplicative subset in , the localization can still be constructed, but the ring homomorphism from to might fail to be injective. For example, if , then is the trivial ring.
- Matsumura (1980), p. 12
- Matsumura (1989), p. 21
- Das, Abhijit; Madhavan, C. E. Veni (2009), Public-key Cryptography: Theory and Practice, Pearson Education India, p. 121, ISBN 9788131708323.
- Hideyuki Matsumura, Commutative algebra, 1980
- Hideyuki Matsumura, Commutative ring theory, 1989