On the divisibility of the discriminant of an integral polynomial by prime powers

The discriminant of an integral polynomial is one of its main characteristics. It influences the distribution of its roots, the structure of the finite extension of the rational field generated by the polynomial's roots. In the paper, we show that, for any given prime power p ^b , there exists an irreducible polynomial with discriminant being a multiple of p ^b .     

Richman extensions of an integral domain

We introduce the notion of a Richman extension of an integral domain, and show its basic properties. We further show that a flat extension need not be a Richman extension, and a Richman extension need not be an LCM-stable extension.     

On a retract of an integral domain

Let A R be commutative integral domains with identity. We say A is a retract of R when there exists a ring homomorphism : RA such that |_A=id_A. We Prove two theorems: Theorem 3.1 says when A is a retract of R via , with Kernel ()= =t·R (O), then t is transcendental over A and if A is sub-inert in R, then RK(t)=A[t], where K is the quotient field of A. This theorem with the hypotheses strengthened to A being a 2-valuation algebra of R and the transcendence degree of R over A equal to one, gives Theorem 4.1, which now says R=A[t]. Given the other hypotheses of Theorem 4.1, A being a 2-valuation algebra of R is equivalent to the existence of a valuation ring V of L, the quotient field of R, such that VR=A. Some counterexamples are given and some variations on the hypotheses of Theorem 4.1 are discussed.     

R+ˆ is surprisingly an integral domain

It is shown that if $(R,\mathfrak{m},k)$ is a complete local domain with $\mathrm{char}k=p>0$ and $R^{+}$ is its integral closure in an algebraic closure of the quotient field, then both the $\mathfrak{m}$-adic and p-adic completions of $R^{+}$ are integral domains. More generally, this theorem remains true if the completeness assumption is relaxed to allow R to be an analytically irreducible Henselian local ring. It is also shown that these rings, which are Cohen-Macaulay R-modules (even balanced in the $\mathfrak{m}$-adic case), will have dimension larger than the dimension of R unless $\dim ?R\le 1$.     

Specialization of integral dependence for modules

We establish the principle of specialization of integral dependence for submodules of finite colength of free modules, as part of the general algebraic-geometric theory of the Buchsbaum–Rim multiplicity. Then we apply the principle to the study of equisingularity of ICIS germs, obtaining results for Whitney’s Condition A and Thom’s Condition A_f. Notably, we describe these equisingularity conditions for analytic families in terms of various numerical invariants, which, for the most part, depend only on the members of a family, not on its total space. Oblatum 18-X-1996 & 16-X-1998 / Published online: 10 June 1999     

A categorification of integral Specht modules

We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories of category for the Lie algebra .

     

Some geometric properties of integral modules

Day's characterization of those spaces 1^p(X_i) which are uniformly convex, in terms of the moduli of convexity of the X_i, is generalized for arbitrary integral modules on measure spaces (K,m) and simplified when m is finite. For this latter purpose a lemma on the moduli of convexity and of smoothness is proved which incidentally gives a further necessary condition for the existence of integral modules in given direct integrals. Further the notions of strict convexity and smoothness of an integral module are related to those of its components.     

On weak-injective modules over integral domains

We show that a weak-injective module over an integral domain need not be pure-injective (Theorem 2.3). Equivalently, a torsion-free Enochs-cotorsion module over an integral domain is not necessarily pure-injective (Corollary 2.4). This solves a well-known open problem in the negative.In addition, we establish a close relation between flat covers and weak-injective envelopes of a module (Theorem 3.1). This yields a method of constructing weak-injective envelopes from flat covers (and vice versa). Similar relation exists between the Enochs-cotorsion envelopes and the weak dimension ?1 covers of modules (Theorem 3.2).     

Krull dimension,overrings and semistar operations of an integral domain

In this paper, we will present new developments in the study of the links between the cardinality of the sets $O(R)$ of all overrings of R, $\mathit{SSF}c(R)$ of all semistar operations of finite character when finite to the Krull dimension of an integral domain R. In particular, we prove that if $|\mathit{SSF}c(R)|=n+\dim R$, then R has at most $n?1$ distinct maximal ideals. Moreover, R has exactly $n?1$ maximal ideals if and only if $n=3$. In this case R is a Prüfer domain with exactly two maximal ideals and Y-graph spectrum. We also give a complete characterizations for local domains R such that $|\mathit{SSF}c(R)|=3+\dim R$, and nonlocal domains R with $|\mathit{SSF}c(R)|=|O(R)|=n+\dim R$ for $n=4$, $n=5$, $n=6$ and $n=7$. Examples to illustrate the scopes and limits of the results are constructed.     

A divisibility nonrelation for the smith invariants of a product of integral matrices

A divisibility formula exists for the Smith invariants of a product of integral matrices that is valid whenever the matrices are diagonal but is otherwise generally invalid     

Vanishing viscosity limit for an expanding domain in space

We study the limiting behavior of viscous incompressible flows when the fluid domain is allowed to expand as the viscosity vanishes. We describe precise conditions under which the limiting flow satisfies the full space Euler equations. The argument is based on truncation and on energy estimates, following the structure of the proof of Kato's criterion for the vanishing viscosity limit. This work complements previous work by the authors, see Iftimie et al. (2009) [5], Kelliher (2008) [8].