MA3412S2_Hil2014.pdf

# If r is an integral domain then the set r of non zero

• Notes
• 38

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If R is an integral domain then the set R * of non-zero elements of R is a multiplicatively closed subset of R . There is thus a corresponding ring R *- 1 R of fractions. The non-zero elements of R *- 1 R are the elements that are of the form s 1 /s 2 , where s 1 , s 2 R * . Each of these elements is a unit of R *- 1 R . It follows that R *- 1 R is a field. In this case the set of non-zero elements of the integral domain R coincides with the set of regular elements of R . It follows that the field R *- 1 R is the total ring of fractions of R . It is also the localization of R at the zero ideal of R . Definition Let R be an integral domain. The field of fractions Frac( R ) of R is the field R *- 1 R , where R * is the set R \ { 0 R } of non-zero elements of R . The basic properties of the field of fractions of an integral domain are summarized in the following results which follow from the discussion above. Proposition 2.44 Let R be an integral domain, and let Frac( R ) be its field of fractions. Then every element of Frac( R ) is represented by a quotient of the form r/s , where r, s R and s 6 = 0 R . Moreover if r , r 0 , s and s 0 are elements of R , and if s 6 = 0 R and s 0 6 = 0 R , then r/s = r 0 /s 0 if and only if s 0 r = sr 0 . The operations of addition and multiplication are defined on the field of fractions Frac( R ) so that ( r 1 /s 1 ) + ( r 2 s 2 ) = ( s 2 r 1 + s 1 r 2 ) / ( s 1 s 2 ) and ( r 1 /s 1 )( r 2 /s 2 ) = ( r 1 r 2 ) / ( s 1 s 2 ) 45

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for all r 1 , r 2 R and s 1 , s 2 R \ { 0 R } . The zero element of Frac( R ) is 0 R / 1 R , and the multiplicative identity element is 1 R / 1 R . The function that sends each element r of the integral domain R to r/ 1 R is an injective unital homomorphism that embeds the integral domain R in its field of fractions. Lemma 2.45 Let R be an integral domain, and let Frac( R ) be the field of fractions of R . Then every unital ring homomorphism from R to a field L extends uniquely to a homomorphism from Frac( R ) to L . Lemma 2.46 Let R be an integral domain, let Frac( R ) be the field of frac- tions of R , and let S be a multiplicatively closed subset of R that does not contain the zero element of R . Then the embedding of R in Frac( R ) induces an embedding of the ring of fractions S - 1 R in the field Frac( R ) . 2.12 Integrally Closed Domains Definition Let R and T be unital commutative rings, where R T . The ring R is said to be integrally closed in T if every element of T that is a root of some monic polynomial with coefficients in R belongs to R . Definition An integral domain R is said to be integrally closed if it is in- tegrally closed in its field of fractions. An integrally closed domain is an integral domain that is integrally closed. Proposition 2.47 All unique factorization domains are integrally closed. Proof Let R be a unique factorization domain, and let r and s be elements of R , where s 6 = 0 R . Suppose that the quotient r/s of r and s in the field of fractions of R is a root of some monic polynomial f ( x ) of degree n with coefficients in R . Then n > 0. Let f ( x ) = a 0 + a 1 x + a 2 x 2 + · · · + a n - 1 x n - 1 + x n .
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• Fall '16
• Jhon Smith
• Algebra, Integers, Prime number, Integral domain, Ring theory, Principal ideal domain

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