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

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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

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

Special Classes of Homogeneous Semilocal Rings Corner Rings

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Basic defjnitions

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Family of Semilocal Rings

Semilocal rings Loca l ring Homogeneous semilocal ringsLocal rings

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Family of Semilocal Rings

Semilocal rings Loca l ring Semiperfect rings Local rings

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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

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Homogeneous semilocal rings Aalgebraic properties

Susan El-Deken Some Properties of Non-commutative Rings of Hurwitz Susan El-Deken Corner Rings 5/31

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Homogeneous semilocal rings Aalgebraic properties

Susan El-Deken Some Properties of Non-commutative Rings of Hurwitz Susan El-Deken Corner Rings 6/31

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Homogeneous semilocal rings

Aalgebraic properties

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Motivation

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Morita Invariant

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Morita Invariant

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Susan El-Deken Some Properties of Non-commutative Rings of Hurwitz

Example

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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].

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Main Results

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Main Results

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Main Results

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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 Series Susan El-Deken Corner Rings 2/31

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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 Series Susan El-Deken Corner Rings 2/31

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Thank you