Near integral domains II

A direct product decomposition is given for the multiplicative semigroup of a finite near integral domain in terms of the subsemigroup of left identities and a group of automorphisms on the additive group of the domain. Conditions are given which insure that every element will have a uniquen-th root. If there existsx≠0 such that (?x)y=?(xy), for eachy, then the additive group of the near integral domain is abelian. Other conditions sufficient for the commutativity of the additive group are given. An example illustrates that non-isomorphic finite near integral domains can have a left ideal decomposition into Sylow subgroups which are isomorphic as near-rings. Another example shows that an infinite near integral domain need not have a nilpotent additive group, even in the d. g. case. It is conjectured that for each natural numbern there is a near integral domain whose additive group is of nilpotent classn.     

M-Canonical ideals in integral domains

We prove that a Priifer domain R has an m-canonical ideal J, that is, an ideal I such that J: (I: J) = J for every ideal J of R, if and only if R is h-local with only finitely many maximal ideals that are not finitely generated; moreover, if these conditions are satisfied, then the product of the non-finitely generated maximal ideals is an m-canonical ideal of R     

Conducive integral domains as pullbacks

This article presents a characterization of conducive (integral) domains as the pullbacks in certain types of Cartesian squares in the category of commutative rings. Such squares behave well with respect to (semi)normalization, thus permitting us to recover the recent characterization by Dobbs-Fedder of seminormal conducive domains. The conducive domains satisfying various finiteness conditions (Noetherian, Archimedean, accp) are characterized by identifying suitable restrictions on the data in the corresponding Cartesian squares. Various necessary or sufficient conditions are given for Mori conducive domains. Consequently, one has examples of several (accp non-Mori; Mori non-Noetherian) conducive domains.Work performed under the auspices of GNSAGA of Consiglio Nazionale delle Ricerche and Dipartimento di Matematica, Università di Roma La SapienzaSupported in part by grants from University of Tennessee Faculty Development Program and the Università di Roma La Sapienza     

Splitting sets in integral domains

Let be an integral domain. A saturated multiplicatively closed subset of is a splitting set if each nonzero may be written as where and for all . We show that if is a splitting set in , then is a splitting set in , a multiplicatively closed subset of , and that is a splitting set in is an lcm splitting set of , i.e., is a splitting set of with the further property that is principal for all and . Several new characterizations and applications of splitting sets are given.

     

Neat submodules over integral domains

Neat subgroups of abelian groups have been generalized to modules in essentially two different ways (corresponding to (a) and (b) in the Introduction); they are in general inequivalent, none implies the other. Here we consider relations between the two versions in the commutative case, and characterize the integral domains in which they coincide: these are the domains whose maximal ideals are invertible.     

Hereditary atomicity in integral domains

If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectively, are hereditarily atomic. As a result, we obtain two classes of hereditarily atomic domains that cannot be embedded into any hereditarily atomic field. By contrast, we show that rings of power series are never hereditarily atomic. Finally, we make some progress on the still open question of whether every subring of a hereditarily atomic domain satisfies ACCP.     

Numerical characterizations of some integral domains

Let R be an integral domain. If R satisfies some finiteness conditions, an algorithm that produces the exact list of all distinct overrings of R is established. Then numerical characterizations of several classes of integral domains are obtained.     

Elasticity of factorizations in integral domains

For an atomic integral domain R, define(R)=sup{mn|x₁x_m=y₁y_n, each x_i,y_jεR is irreducible}. We investigate (R), with emphasis for Krull domains R. When R is a Krull domain, we determine lower and upper bounds for (R); in particular,(R)≤max{|Cl(R)| 2, 1}. Moreover, we show that for any real numbers r≥1 or R=∞, there is a Dedekind domain R with torsion class group such that (R)=r.     

Fragmented domains have infinite Krull dimension

Résumé  On dit qu'un anneau intègreR est fragmenté si pour tout élément non-inversibler deR, il existe un élément non-inversibles deR tel que r∈∩Rs ⁿ. On montre, pour un anneau intègreR qui n'est pas un corps, qu'il existe un idéal maximal deR qui contient une cha?ne strictement croissante d'idéaux premiers deR. Si, de plus,R n'ax qu'un nombre fini d'idéaux maximaux, alors on peut reformuler l'affirmation précédente pour tout idéal maximal deR. Il découle que toute anneau intègreR, qui n'est pas un corps et qui possède un idéal premierP tel queR+PR _p soit fragmenté, doit être de dimension infinie (au sens de Krull). On donne un exemple d'un tel anneauR qui n'est pas fragmenté.      

The w-integral closure of integral domains

Let D be an integral domain with quotient field K, $\overline{D}$ the integral closure of D, X an indeterminate over D, and $N_v=\{f\in D[X]|{(A_f)}_v=D\}$. Let w be the 1-operation on D defined by $I_w=\{x\in K|\text{there is a finitely generated ideal}A\text{such that}{A}^{\mathrm{-1}}=D\text{and}xA\subseteq I\}$, and let $D^w=\{u\in K|u{I}_w\subseteq I_w\text{for some nonzero finitely generated ideal}I\text{of}D\}$. Then $D^w$, called the w-integral closure of D, is an integrally closed overring of D. In this paper, we show that $D^w=\overline{D}{[X]}_{N_v}\cap K$ and $D^w{[X]}_{N_v}=\overline{D}{[X]}_{N_v}$. Using this result, we give several w-integral closure analogs of the integral closure. We also study the w-integral closure of UMT-domains and strong Mori domains.