A NOTE ON COMMUTATIVE RINGS WITH ZERO DIVISORS

Abstract

Let R be a commutative ring with identity and no nonzero nilpotents, that is R is a reduced ring. In this thesis, we show the following results: (1) If R is a integrally closed ring for which its total quotient ring T(R) is strongly Pru¨fer, then R is integrally closed in Q_(0)(R), the ring of finite fractions of R. (2) R[X] is completely integrally closed if and only if R is completely integrally closed in Q_(0)(R).;R을 항등원을 갖고 0이 아닌 nilpotent를 갖지 않는 환이라 하자. 이 논문에서 우리는 다음을 보인다: (1) 만약 R이 integrally closed환이고 분수환T(R)이 strongly Prufer라면 R이 유한분수환Q_(0)(R)에서 integrally closed가 된다. (2) R[X]가 completely integrally closed가 되기 위한 필요 충분 조건은 R이 Q_(0)(R)에서 completely integrally closed가 되는 것이다.