Two star-operations and their induced lattices

Let D be an integral domain with quotient field K, let F(D) (f(D)) be the s et of nonzero (finitely generated) fractional ideals of D, and let * be a s tar-operation on F(D). For A epsilon F(D) define A (*) over bar = {x epsilo n K \ there exists a J epsilon F(D) such that J* = D and xJ subset of or eq ual to A} and A(w)(*) = {x epsilon K \ there exists J epsilon f(D) such tha t J* = D, and xJ subset of or equal to A}. Then (*) over bar and *(w), are star-operations on F(D) that satisfy (A boolean AND B)(*) over bar = A (*) over bar boolean AND B (*) over bar and (A boolean AND B)*(w) = A*(w) boole an AND B*(w). Moreover, (*) over bar (*(w)) is the greatest (finite charact er) star-operation Delta subset of or equal to * with (A boolean AND B)(Del ta) = A(Delta) boolean AND B-Delta. We also show that *(w)-Max(D) = *(s)-Ma x(D) and A*(w) = boolean AND{A(p) \ P epsilon *(s)-Max(D)}. Let L*(w)(D) = {A \ A is an integral *(w)-ideal}boolean OR{0}. Then L*(w)(D) forms an r-la ttice. If D satisfies ACC on integral *w-ideals, L*(w)(D) is a Noether latt ice and hence primary decomposition, the Krull intersection theorem, and th e principal ideal theorem hold for *(w)-ideals of D. For the case of * = up silon, *(w) is the w-operation introduced by Wang Fanggui and R.L. McCaslan d.