A note on *_{𝑤}-Noetherian domains

Abstract

Let D be an integral domain with quotient field K, * be a staroperation on D, and GV * (D) be the set of finitely generated ideals J of D such that J * = D. Then the map * w defined by I * w = {x ∈ K | Jx ⊆ I for some J ∈ GV * (D)} for all nonzero fractional ideals I of D is a finite character staroperation on D. In this paper, we study several properties of * w -Noetherian domains. In particular, we prove the Hilbert basis theorem for * w -Noetherian domains.

The Strong Mori Property in Rings With Zero Divisors

“…If not, J 0 ⊆ P α for some P α ∈ P. Then P −1 α ⊆ J −1 0 = D. Hence P −1 α = D, which is a contradiction to the choice of P α , so we have J = J 0 R = J 0 (+)B is a finitely generated ideal of R and J = J 0 R = J 0 (+)B is a finitely generated semiregular ideal of R by (2). By (3), Hom R (J, R) [5,Theorem 4.4], so R satisfies the restricted DCC on semiregular v-ideals.…”

“…. }, where each X i is an indeterminate over K. Let [2] , X [3] , Y F , ZF ] and P denote the set of primes of D which do not contain both Y and Z. Let R be the A + B ring corresponding to D and P(see [1,Section 26]).…”

A SIMPLE PROOF OF HILBERT BASIS THEOREM FOR *ω-NOETHERIAN DOMAINS

“…One of the most beautiful results in commutative algebra is Hilbert basis theorem, which states that if D is a Noetherian domain, then so is the polynomial ring D [X]. In [4,Theorem 3.2], the authors generalized and proved the * w -Noetherian domain version of Hilbert basis theorem, which mentions that if * is a star-operation on D[X], then D being a * w -Noetherian domain implies D[X] being a * w -Noetherian domain.…”

Integral Domains in Which Every Nonzero w-Flat Ideal Is w-Invertible

“…. , f m )D M [X][7] (Theorem 4.3) (or[14] (Proposition 2.10)); so ID M [X] = X n ID M [X] + ( f 1 , . .…”