**Irreductibility,
Factorizations, Krull Dimension, and their Computational Aspects in**

The project objectives are the following 7 topics:

**1. Irreducible
decompositions in lattices with applications to torsion theories and
Grothendieck categories.** First,
we intend to extend from modules to upper continuous modular lattices
the main result of Fort [Fo 67] concerning the characterization of
modules M rich in coirreducibles by means of irredundant irreducible
decompositions of 0 in any submodule of M. Then, we consider a
similar problem by replacing coirreducible submodules with
subdirectly irreducible submodules. We expect that the lattices
having this property, we call lattices rich in completely
coirreducibles, are exactly the atomic lattices. Then, we will apply
these lattice-theoretical results obtained to Grothendieck categories
and module categories equipped with a hereditary torsion theory.

**2. Primal, completely
irreducible, and primary decompositions in modules over arbitrary
commutative rings. **As the
Lasker-Noether Theorem has been extended from Noetherian rings to
Noetherian modules cf. [Bo 89], a next goal is to extend from ideals
to modules over commutative rings some results of L. Fuchs, W.
Heinzer, B. Olberding [FuHeOl 04], [FuHeOl 06], and W. Heinzer, B.
Olberding [HeOl 05] concerning primal and completely irreducible
ideals, to investigate their links with primary decompositions, as
well as the uniqueness and irredundancy of such decompositions.

**3. Krull dimension,
dual Krull dimension, and the Faith’ s theorem.**

(i*)
The relationship between Krull dimension and dual Krull dimension of
a ring.* An open
question raised by Albu and Smith in 1993 asks whether the dual Krull
dimension of any (commutative) ring R having Krull dimension is
bounded above by its Krull dimension. Partial results concerning this
question have been obtained by Albu and Smith ([ASm 95], [ASm 99]).
We intend to deal with this problem as well as with its
relativization with respect to a hereditary torsion theory.

(ii)
*Faith's Theorem.*
A nice result of *C.
Faith* [Fa 99]
establishes that a right module M is Noetherian if and only if M is
QFD (i.e., quotient (Goldie) finite dimensional) and satisfies the
ascending chain condition on subdirectly irreducible submodules.
Faith's result can be reformulated in a dual Krull dimension setting
for the lattice L(M) of all submodules of M and its subset S(M) of
subdirectly irreducible submodules of M as follows: the dual Krull
dimension of L(M) is at most 0 if and only if L(M) is a QFD lattice
and the dual Krull dimension of the poset S(M) is at most 0. A
natural, still open, question raised by Albu and Rizvi in [ARi 01]
asks whether the above reformulation of Faith's result holds for an
arbitrary ordinal alpha instead of 0. A positive answer to this
question for any finite ordinal in given in Albu, Iosif, Teply [AIoTe
05]. We expect to prove the result for an arbitrary transfinite
ordinal.

**4. Heights and
Bombieri’s norms estimates for polynomials factors****.**
We will refine some inequalities
concerning the height and the Bombieri’s norm of the divisors
of a univariate complex polynomial. For polynomials with integer
coefficients we will obtain sharper estimates than those given by the
inequalities of Landau and Beauzamy.

**5. Bounds for the
absolute values and the multiplicities of the polynomial roots.**
We will obtain new results on the
root location for real and complex polynomials. We will obtain upper
bounds for the multiplicities of the roots, in particular
separability criteria.

**6.
****Applications of Newton’s polygon method to
polynomial factorization.** We
will use the properties of the slopes in the Newton polygon
associated to a product of polynomials, in the study of the
factorization of multivariate polynomials over an algebraically
closed field. In particular, some new irreducibility criteria will be
obtained.

**7. ****The
study of the irreducibility of linear combinations of relatively
prime polynomials****,
Cogalois groups and the Hardy-Littlewood conjecture.**

We will study the irreducibility of some classes of univariate and multivariate polynomials that can be expressed as linear combinations of relatively prime polynomials. In particular we will obtain new irreducibility conditions for Schoenemann–like polynomials, we will study the Cogalois groups for various classes of polynomials. We will also investigate some algebraic - analytical and computational aspects of the Hardy-Littlewood conjecture.

**REFERENCES**

[AIoTe
04] T. ALBU, M. IOSIF, M.L. TEPLY, *Modular QFD lattices with
applications to Grothendieck categories and torsion theories, *J.
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[AIoTe
05] T. ALBU, M. IOSIF, M.L. TEPLY, *Dual Krull dimension and
quotient finite dimensionality*, J. Algebra 284 (2005) , 52-79.

[AKrTe
00] T. ALBU, G. KRAUSE, M.L. TEPLY, *The nilpotence of the
tau-closed prime radical in rings with tau-Krull dimension*, J.
Algebra 229 (2000), 498-513.

[AKrTe
01] T. ALBU, G. KRAUSE, M.L. TEPLY, *Bijective relative Gabriel
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[ASm
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[ASm
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[ASm
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[ASm
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[Bon
04] N.C. BONCIOCAT, *Upper bounds for the number of factors for a
class of polynomials with rational coefficients*, Acta Arith. 113
(2) (2004), 175-187.

[Bon
07] N.C. BONCIOCAT, *Irreducibility criteria for pairs of
polynomials whose resultant is a prime number*, submitted.

[BoBo
06a] N.C. BONCIOCAT, A.I. BONCIOCAT, *Some classes of irreducible
polynomials*, Acta Arith. 123 (2006), 349-360.

[BoBo
06b] N.C. BONCIOCAT, A.I. BONCIOCAT, *The irreducibility of
polynomials that have one large coefficient and take a prime value*,
to appear in Canad. Math. Bull.

[BoBo
06c] A.I. BONCIOCAT, N.C. BONCIOCAT, *A Capelli type theorem for
multiplicative convolutions of polynomials*, to appear in Math.
Nachr.

[BoBoZa
06] N.C. BONCIOCAT, A.I. BONCIOCAT, A. ZAHARESCU, *Bounds for the
multiplicities of the roots for some classes of complex polynomials,*
Math. Inequal. Appl. 9 (1) (2006), 11-22.

[BoBoZa
07] N.C. BONCIOCAT, A.I. BONCIOCAT, A. ZAHARESCU, *On the number of
factors of convolutions of polynomials with integer coefficients,*
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[Cav
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[CaVaZa
02] M. CAVACHI, M. VAJAITU, A. ZAHARESCU, *A class of irreducible
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02] M. FIILASETA, T.-Y. LAM, *On the irreducibility of the
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Albu Toma, Senior Researcher I in the Institute of Mathematics "Simion Stoilow" of the Romanian Academy

Doru Stefanescu

Alexandru Gica

Nicolae Ciprian Bonciocat

Vlad Copil – Ph. D student

T. Albu & J. Van Den Berg, "An indecomposable non-locally finitely generated Grothendieck category with simple objects, J. Algebra 321(2009), 1358-1545.

T. Albu, "Completely irreducible meet decompositions in lattices, with applications to Grothendieck categories and torsion theories (I)", Bull. Math. Soc. Sci. Math. Roumanie 52(100) (2009), 393-419.

T. Albu, "Completely irreducible meet decompositions in lattices, with applications to Grothendieck categories and torsion theories (II)", Bull. Math. Soc. Sci. Math. Roumanie 53(101) (2010), to appear.

`D. Stefanescu: "Inequalities on real roots of polynomials", to appear in MIA (Mathematical Inequalities and Applications), no. 4/2009.`