GEBZE INSTITUTE OF TECHNOLOGY (GIT)
DEPARTMENT OF MATHEMATICS
GIT MATHEMATICS SEMINARS
” Commutativity conditions on generalized
derivations in rings with involution ”
1
by
Cihat Abdioğlu
Abstract:
Let (R;*) be a ring with involution, I be an ideal of R ,and let Z(R) be the center of R.
The purpose of this talk is to explore the commutativity of R if it admits a generalized
derivation F satisfying one of the following conditions:
(1) F(xy) – xy is an element of Z(R) for all elements x, y of R,
(2) F(xy) - yx is an element of Z(R) for all elements x, y of R,
(3) F(xy) – xy is an element of Z(R) for all elements x, y of I,
(4) F(xy) - yx is an element of Z(R) for all elements x, y of I.
Date :
June 1, 2010
Time : 13:30
Place : Seminar Hall, Mathematics Department2
1
Joint paper with Serap Şahinkaya
2
Adres : Gebze İleri teknoloji Enstitüsü, Matematik Bölümü, H-Blok, Çayırova Kampüsü, 41400
Gebze, Kocaeli.
GEBZE INSTITUTE OF TECHNOLOGY (GIT)
DEPARTMENT OF MATHEMATICS
GIT MATHEMATICS SEMINARS
” Regularity of direct sums decompositions of modules”
by
Şule Ecevit
Absract:
I will present a geometrical descriptions of the behavior of direct sum decompositions of modules. This
regular geometrical behavior has been discovered for modules with a semilocal endomorphism ring in
general [3], for uniserial modules [2], kernels of morphisms between indecomposable injective modules
[5] and cyclically presented modules [1] and [4]. In [1], the authors obtain a surprising analogy between the
behavior of direct sums of uniserial modules over arbitrary ring and the behavior of direct sums of
cyclically presented modules over local rings. This research is based on the observation that both the
endomorphism ring of a uniserial module over an arbitrary ring and the endomorphism ring of a cyclically
presented module over a local ring are either local or exactly have two maximal ideals whose reside rings
are division rings.
We will present what happens for an arbitrary cyclically finitely presented modules
dimension < 1 or =1 (Is this sufficient for the weak Krull-Schmidt Theorem to hold?).
of projective
[1] B. Amini, A. Amini and A. Facchini, Equivalence of diagonal matrices over local rings, J. Algebra 320
(2008), 1288-1310.
[2] A. Facchini, Krull-Schmidt fails for serial modules, Trans. Amer. Math. Soc. 348 (1996), 4561--4575.
[3] A. Facchini, Direct sum decompositions of modules, semilocal endomorphism rings, and Krull
monoids, J. Algebra, 256 (2002), 280--3007.
[4] A. Facchini and N. Girardi, Couniformly presented modules and dualities, to appear in the
Proceedings of the ``Conference on Algebra and its Applications'' in honor of 70th Birthday of S. K. Jain,
Dinh Van Huynh and Sergio R. L\'opez Permouth Eds., Birkh\"auser Verlag, Basel, Heidelberg, London,
New York, 2009.
[5] Ş. Ecevit, A. Facchini, M. T. Koşan, Kernels of morphisms between indecomposable injective
modules, to appear in Glasgow Math. J., 2010.
Date :
June 1, 2010
Time : 13:00
Place : Seminar Hall, Mathematics Department2