Abstract
In § 10 (Example 3) of the preceding chapter it was briefly indicated how the well-known construction of the field of fractions of a commutative integral domain could be generalized to arbitrary commutative rings. When trying to extend this construction to non-commutative rings, one finds that this is not always possible, but that one can give necessary and sufficient conditions for the existence of a ring of fractions. Such a condition was first found by Ø. Ore [1] around 1930 for the case of a skew-field of fractions of a domain. The existence of a total ring of fractions of an arbitrary ring was first considered by K. Asano [1]. General rings of fractions were studied by Elizarov [1], and a systematic theory of rings and modules of fractions was developed by P. Gabriel [1,2] in connection with his theory of general rings of quotients.
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© 1975 Springer-Verlag Berlin Heidelberg
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Stenström, B. (1975). Rings of Fractions. In: Rings of Quotients. Die Grundlehren der mathematischen Wissenschaften, vol 217. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-66066-5_4
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DOI: https://doi.org/10.1007/978-3-642-66066-5_4
Publisher Name: Springer, Berlin, Heidelberg
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