Research ArticleOpen AccessGoogle Scholar indexed
NC-Rings and Some Commutativity Conditions
Department of Mathematics, King Abdulaziz University, Jeddah, KSA
Department of Mathematics, King Abdulaziz University, Jeddah, KSA
- 1 Department of Mathematics, King Abdulaziz University, Jeddah, KSA
- 2 Department of Mathematics, King Abdulaziz University, Jeddah, KSA
Advances in Pure Mathematics·Volume 09 (2019)·Pages 143–163·Published 14 February 2019·DOI10.4236/apm.2019.92008
Copy link · social · email
Abstract
Sum of two nilpotent elements in a ring may not be nilpotent in general, but for commutative rings this sum is nilpotent. In between commutative and non-commutative rings there are several types of rings in which this property holds. For instance, reduced, NI, AI (or IFP), 2-primal, reversible and symmetric, etc. We may term these types of rings as nearby commutative rings (in short NC-rings). In this work we have studied properties and various characterizations of such rings as well as rngs. As applications, we have investigated some commutativity conditions by involving semi-projective-Morita-contexts and right C k -Goldie rings.
KeywordsNC-RingsNC-RngsSemi-Projective-Morita-ContextsRight C<sub>k</sub>-Goldie Rings
- Jacobson, N. (1980) Basic Algebra-II. W. H. Freeman & Company, San Francisco.
- Lambek, J. (1971) On the Representation of Modules by Sheaves of Factor Modules. Canadian Mathematical Bulletin, 14, 359-368. https://doi.org/10.4153/CMB-1971-065-1
- Lam; T.Y. (1999) Lectures on Modules and Rings. Graduate Texts in Math., Vol. 189, Springer-Verlag, New-York. https://doi.org/10.1007/978-1-4612-0525-8
- Morita, K. (1958) Duality for Modules and Its Applications to the Theory of Rings with Minimum Conditions. Science Reports of the Tokyo Kyoiku Daigaku, 6A, 83-142.
- Nauman, S.K. (1994) Intersecting Subcategories of Static Modules, Stable Clifford Theory and Colocalization-Localization. Journal of Algebra, 170, 400-421. https://doi.org/10.1006/jabr.1994.1344
- Nauman, S.K. (2004) Morita Similar Matrix Rings and Their Grothendieck Groups. The Alig. Bull. Math, 23, 49-60.
- Al-Kenani, A.N. and Nauman, S.K. (2008) A Short Construction of Morita Similar Matrix Rings. JP Journal of Algebra, Number Theory, and Applications, 11, 203-207.
- Birkenmeier, G.F., Heatherly, H.E. and Lee, E.K. (1993) Completely Prime Ideals and Associated Radicals, Ring Theory (Granville, OH 1992). World Scientific Publisher, River Edge, NJ, 102-129.
- Marks, G. (2003) A Taxonomy of 2-Primal Rings. Journal of Algebra, 266, 494-520. https://doi.org/10.1016/S0021-8693(03)00301-6
- Marks, G. (2001) On 2-Primal Ore Extensions. Communications in Algebra, 29, 2113-2123. https://doi.org/10.1081/AGB-100002173
- Hwang, S.U., Jeon, Y.C. and Park, K.G. (2007) On NCI Rings. Bulletin of the Korean Mathematical Society, 44, 215-223. https://doi.org/10.4134/BKMS.2007.44.2.215
- Bell, H.E. (1970) Near Rings in Which Each Element Is a Power of Itself. Bulletin of the Australian Mathematical Society, 2, 363-368. https://doi.org/10.1017/S0004972700042052
- Ham, K., Jeon, Y., Kang, J., Kim, N., Lee, W., Lee, Y., Ryu, S. and Yang, H. (2008) IFP Rings and Near-IFP Rings. Journal of the Korean Mathematical Society, 45, 727-740. https://doi.org/10.4134/JKMS.2008.45.3.727
- Jeon, Y.C., Kim, H.K., Lee, Y. and Yoon, J.S. (2009) On Weak Armendariz Rings. Bulletin of the Korean Mathematical Society, 46, 135-136. https://doi.org/10.4134/BKMS.2009.46.1.135
- Marks, G. (2002) Reversible and Symmetric Rings. Journal of Pure and Applied Algebra, 174, 311-318. https://doi.org/10.1016/S0022-4049(02)00070-1