show that every nonzero element of a integral domain R is a

show that every nonzero element of a integral domain R is a unit in a field F of quotients of R

Solution

Let an integral domain be R, we have to show that a field F satisfies the field of Quotients in a unit R.

let the field F = {a,b..} where aR and bR be a non-zero element.

by cancellation law, {(a,b)~(c,d)} implies that ad = bc.then,

for {(a,b),(c,d)} F ,

{( a,b) + (c,d)} = {(ad + bc,bd)}

{( a,b) . (c,d)} = {(ac ,bd)}

which shows the addition and multiplication operations.

from that a field F satisfies the field of Quotients in a unit R.