Special Classes of Homogeneous Semilocal Rings Corner Rings Susan - PowerPoint PPT Presentation

Special Classes of Homogeneous Semilocal Rings Corner Rings Susan F. El-Deken Mathematics Department Faculty of Science Helwan University, Cairo, Egypt Groups, Rings, Associated Structures Spa, Belgium, 13 June, 2019

Basic defjnitions Susan El-Deken Corner Rings 3/31

Family of Semilocal Rings Semilocal rings Loca l Homogeneous ring semilocal ringsLocal rings

Family of Semilocal Rings Semilocal rings Loca Semiperfect l rings ring Local rings

Maximal two-sided ideal In non-commutative ring, local ring have a unique maximal left ideal (unique maximal right ideal) equivalent to have a unique maximal two-sided ideal but if the ring having a unique maximal two sided ideal is not equivalent to being local. An extension class of local ring, which has a unique maximal two-sided ideal, is called homogeneous semilocal ring. The Jacobson radical of a homogeneous semilocal ring is its unique maximal two-sided ideal S usan El-Deken Corner Rings 4/31

Homogeneous semilocal rings A algebraic properties Susan El-Deken Corner Rings 5/31 Susan El-Deken Some Properties of Non-commutative Rings of Hurwitz

Homogeneous semilocal rings A algebraic properties Susan El-Deken Corner Rings 6/31 Susan El-Deken Some Properties of Non-commutative Rings of Hurwitz

Homogeneous semilocal rings Aalgebraic properties

Motivation Susan El-Deken Corner Rings 7/31

Morita Invariant Susan El-Deken Corner Rings 8/31

Morita Invariant Susan El-Deken Corner Rings 9/31

Susan El-Deken Corner Rings 10/31

Susan El-Deken Corner Rings 11/31

Example Susan El-Deken Corner Rings 12/31 Susan El-Deken Some Properties of Non-commutative Rings of Hurwitz

Question Now we raise the following question: Under what conditions, would a ring with homogeneous semilocal corner rings be a homogeneous semilocal ring? Before answer this question, we quote the following useful result from [34]. Susan El-Deken Corner Rings 13/31

Main Results

Main Results

Main Results

Reference [1] R. Corisello and A. Facchini, Homogeneous semilocal rings, Comm. Algebra 29(4) (2001) 1807–1819. [2] A. Facchini, Module Theory: Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progress in Math., Vol. 167, Birkhauser Boston, 1998. [3] M.H. Fahmy, S.F. El-Deken, S.M. Abdelwahab, Normalizing extensions of homogeneous semilocal rings and related rings, Journal of the Egyptian Mathematical Society 20(2012) 50–52. [4] ,Homogeneous semilocal group rings and crossed products, Journal of Algebra and Its Applications, 11(6) (2012). Susan El-Deken Some Properties of Non-commutative Rings of Hurwitz Susan El-Deken Corner Rings 2/31 Series

Reference [5] T.Y.Lam, A First Course in Non-Commutative Rings, GTM 131, Springer, Berlin, 1991. [6] T. Y. Lam, Exercises in Classical Ring Theory, Springer, Berlin, 1995 [7] J. Lambek, Lectures on rings and modules, Blaisdell, London, 1966. [8] Y. Lee and C. Huh, On rings in which every maximal one-sided ideal contains a maximal ideal, Comm. Algebra 27(8) (1999) 3969–3978. Susan El-Deken Some Properties of Non-commutative Rings of Hurwitz Susan El-Deken Corner Rings 2/31 Series

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