The difference between Field of fractions and Field of quotients
When used as nouns, field of fractions means the smallest field in which a given ring can be embedded, whereas field of quotients means a field all of whose elements can be represented as ordered pairs each of whose components belong to a given integral domain, such that the second component is non-zero, and so that the additive operator is defined like so: (a,b) + (a',b') = (a b' + a' b,b b'), the multiplicative operator is defined coordinate-wise, the zero is (0,1), the unity is (1,1), the additive inverse of (a,b) is (-a,b), equivalence is defined like so: (a,b) \equiv (a', b') if and only if a b' = a' b, and multiplicative inverse of a non-zero–equivalent element (a,b) is (b,a).
check bellow for the other definitions of Field of fractions and Field of quotients
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Field of fractions as a noun (algebra, ring theory):
The smallest field in which a given ring can be embedded.
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Field of quotients as a noun (algebra):
A field all of whose elements can be represented as ordered pairs each of whose components belong to a given integral domain, such that the second component is non-zero, and so that the additive operator is defined like so: (a,b) + (a',b') = (a b' + a' b,b b'), the multiplicative operator is defined coordinate-wise, the zero is (0,1), the unity is (1,1), the additive inverse of (a,b) is (-a,b), equivalence is defined like so: (a,b) \equiv (a', b') if and only if a b' = a' b, and multiplicative inverse of a non-zero–equivalent element (a,b) is (b,a).