DUALS OF IDEALS IN RINGS WITH ZERO DIVISORS

摘要

For a nonzero ideal I of R, we defineI~(-1) = (R : I) = {x ∈ Q(R)|xI ∩ R} and call it the dual of I where Q(R) is complete ring of quotients of R. Much work has been with regard to determining when (R : I) is a ring in the case R is an integral domain. This paper will extend those results to dense ideals in rings with zero divisors. We will prove several properties with duals of prime ideals including for a dense prime P of ring R, (R : P)≠ (P : P) if and only if PRp is invertible and P is of the form P (R: (1, x)) for some x ∈ Q(R). Attention will also be given to duals of ideals in Prufer and strong Prufer rings. Such as if P is a semiregular prime ideal of strong Prufer ring R and P is noninvertible then P~(-1) = (P : P) is a ring.