Journal article
Ideal-Symmetric and Semiprime Rings
Communications in Algebra, Vol.41(12), pp.4504-4519
12/02/2013
DOI: 10.1080/00927872.2012.705402
Abstract
Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/x n R[x] is ideal-symmetric over a semiprime ring R.
Details
- Title: Subtitle
- Ideal-Symmetric and Semiprime Rings
- Creators
- Victor Camillo - Department of Mathematics , The University of Iowa, Iowa CityTai Keun Kwak - Department of Mathematics , Daejin UniversityYang Lee - Pusan National University
- Resource Type
- Journal article
- Publication Details
- Communications in Algebra, Vol.41(12), pp.4504-4519
- Publisher
- Taylor & Francis Group
- DOI
- 10.1080/00927872.2012.705402
- ISSN
- 0092-7872
- eISSN
- 1532-4125
- Language
- English
- Date published
- 12/02/2013
- Academic Unit
- Mathematics
- Record Identifier
- 9983985702002771
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