Tiberiu Dumitrescu

S-NOETHERIAN RINGS

ABSTRACT A commutative ring R with identity is called S-Noetherian, where is a given multiplicative set, if for each ideal I of R, for some and some finitely generated ideal J. Using this concept, we tie together several different known results. For instance, the fact that is a Noetherian ring whenever R is so, and that is Noetherian whenever for each nonzero element d of the domain D areboth consequences of the following result: If R is an S-Noetherian ring, then so is , provided for each where S consists of nonzerodivisors.

Almost Splitting Sets and AGCD Domains

Abstract Let D be an integral domain. A multiplicative set S of D is an almost splitting set if for each 0 ≠ d ∈ D, there exists an n = n(d) with d n = st where s ∈ S and t is v-coprime to each element of S. An integral domain D is an almost GCD (AGCD) domain if for every x, y ∈ D, there exists a positive integer n = n(x, y) such that x n D ∩ y n D is a principal ideal. We prove that the polynomial ring D[X] is an AGCD domain if and only if D is an AGCD domain and D[X] ⊆ D′[X] is a root extension, where D′ is the integral closure of D. We also show that D + XD S [X] is an AGCD domain if and only if D and D S [X] are AGCD domains and S is an almost splitting set.

Quasi-Schreier Domains II

We study a class of integral domains characterized by the property that every nonzero finite intersection of principal ideals is a directed union of invertible ideals.

t-Schreier Domains

We study, under the name t-Schreier, the class of those integral domains whose group of t-invertible t-ideals satisfies the Riesz interpolation property. The so-called Prüfer v-multiplication domains (PVMDs) and Prüfer domains are special cases of t-Schreier domains. We show that, while a number of results known for Prüfer domains and PVMDs hold for these domains, the t-Schreier domains have a remarkable capability of unifying various results in that the results proved for t-Schreier domains can also be translated to results on pre-Schreier domains and hence on GCD and Bezout domains.

Almost-Schreier Domains

As an extension of the class of (pre)-Schreier domains introduced by P. M. Cohn and M. Zafrullah, we introduce and study a class of integral domains D characterized by the property that whenever a, b 1, b 2 ∈ D − {0} and a | b 1 b 2, there exist an integer k ≥ 1 and a 1, a 2 ∈ D − {0} such that a k = a 1 a 2 and , i = 1, 2. We call them almost-Schreier domains. We show that an almost-Schreier domain has torsion t-class group, that a local (Noetherian) one-dimensional domain is almost-Schreier and that the polynomial ring with coefficients in an integrally closed almost-Schreier domain is almost-Schreier.

Regularity and finite flat dimension in characteristic p>0

To a homomorphism u: A → B of Noetherian rings of positive prime characteristic p, we associate the homomorphism given by for each a where ) denotes the A-algebra structure on A obtained via the r-th power of the Rrobenius endomorphism of A. If u is fiat, we show that u is regular if and only if has finite flat dimension if and only if is flat. We also give sufficient conditions for the regularity of a homomorphism u with of finite flat dimension. In the spirit of [A2, Theoreme 23] and [R2, Corollaire 5] we obtain new characterizations for the G-rings of characteristic p. We also state some criteria of Noetherianity for the rings A p ⊗ A B (A B Noetherian rings) and (A fieldB Noetherian ring).

LCM-splitting sets in some ring extensions

Let S be a saturated multiplicative set of an integral domain D. Call S an lcm splitting set if dD s n D and dD n sD are principal ideals for every d E D and s ∈ S. We show that if R is an R 2 -stable overring of D (that is, if whenever a,b ∈ D and aD ∩bD is principal, it follows that (a.D∩bD)R = aR∩bR) and if S is an lcm splitting set of D, then the saturation of S in R is an lcm splitting set in R. Consequently, if D is Noetherian and p ∈ D is a (nonzero) prime element, then p is also a prime element of the integral closure of D. Also, if D is Noetherian, S is generated by prime elements of D and if the integral closure of D s is a UFD, then so is the integral closure of D.

Domains whose overrings satisfy accp

An integral domain D satisfies ACC on principal ideals (ACJCP) if there does not exist an infinite strictly ascending chain of principal ideals of D. Any Noetherian domain, in particular any Dedekind domain, satisfies ACCP. In this note we prove the following theorem: Let D be an integral domain. Then the integral closure of D is a Dedekind domain if and only if every overring of D (ring between D and its quotient field) satisfies ACCP.

A Schreier Domain Type Condition II

For an integral domain D and a star operation * on D, we study the following condition: whenever I>AB with I, A, B nonzero ideals, there exist nonzero ideals H and J such that I*=(HJ)*, H*>A and J*>B.

Almost Quasi-Schreier Domains

As an extension of the (pre)-Schreier domains studied by Cohn, McAdam, Rush, and Zafrullah, we study the class of integral domains D characterized by the property that whenever I ⊇ J 1 J 2 with I, J 1, J 2 invertible ideals of D, there exist an integer k ≥ 1 and ideals I 1, I 2 such that I k = I 1 I 2, , . The quasi-Schreier domains and the almost-Schreier domains, recently introduced by the second author, Moldovan and Khalid, satisfy this property.