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## CHAPTER 12

### Rings of fractions

Let R be a commutative ring containing regular elements; that is, elements such that a ≠ 0 and a is not a zero divisor. In this chapter we show that any commutative ring R with regular elements can be embedded in a ring Q with unity such that every regular element of R is invertible in Q. In particular, any integral domain can be embedded in a field. Indeed, by defining the general notion of ring of fractions with respect to a multiplicative subset S, we obtain a ring Rs such that there is a canonical homomorphism from R to Rs. The conditions under which a noncommutative integral domain can be embedded in a division ...

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