Commutative Ring Theory

18 The Ring of Finite Fractions

THOMAS G. LUCAS Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina

1. Introduction

Let R be a commutative ring with unit having total quotient ring T(R). The total quotient ring of R is made up of the “simple” fractions of the form r/s where both r and s are in R and s is regular; i.e., s is not a zero divisor. (As in forming the quotient field of an integral domain, two fractionstic> is not a zero divisor. (As in forming the quotient field of an integral domain, two fractions r/s and t/u are equivalent if $ru=ts$.) The fraction r/s can also be viewed as the R-module homomorphism from the principal ideal sR into R which maps s to r. Of course for a nonzero ideal