Abstract
References
- L. V. An, T. G. Nam and N. S. Tung, On quasi-morphic rings and related problems, Southeast Asian Bull. Math., 40(1) (2016), 23-34.
- V. Camillo and W. K. Nicholson, Quasi-morphic rings, J. Algebra Appl., 6(5) (2007), 789-799.
- V. Camillo, W. K. Nicholson and Z. Wang, Left quasi-morphic rings, J. Algebra Appl., 7(6) (2008), 725-733.
- H. Chen, Regular elements in quasi-morphic rings, Comm. Algebra, 39 (2011), 1356-1364.
- A. J. Diesl, T. J. Dorsey and W. W. McGovern, A characterization of certain morphic trivial extensions, J. Algebra Appl., 10(4) (2011), 623-642.
- G. Ehrlich, Units and one-sided units in regular rings, Trans. Amer. Math. Soc., 216 (1976), 81-90.
- K. R. Goodearl, Von-Neumann Regular Rings, Pitman, Boston, 1979.
- Q. Huang and J. Chen, $\pi$-morphic rings, Kyungpook Math. J., 47(3) (2007), 363-372.
- T. K. Lee and Y. Zhou, Morphic rings and unit regular rings, J. Pure Appl. Algebra, 210 (2007), 501-510.
- T. K. Lee and Y. Zhou, Regularity and morphic property of rings, J. Algebra, 322 (2009), 1072-1085.
- T. K. Lee and Y. Zhou, A theorem on unit regular rings, Canad. Math. Bull., 53(2) (2010), 321-326.
- W. K. Nicholson and E. S. Campos, Rings with the dual of the isomorphism theorem, J. Algebra, 271 (2004), 391-406.
Abstract
The main objective of this paper is to study (quasi-)morphic property of skew polynomial rings. Let $R$ be a ring, $\sigma$ be a ring homomorphism on $R$ and $n\geq 1$. We show that $R$ inherits the quasi-morphic property from $R[x;\sigma]/(x^{n+1})$.
It is also proved that the morphic property over $R[x;\sigma]/(x^{n+1})$ implies that $R$ is a regular ring. Moreover, we characterize a unit-regular ring $R$ via the morphic property of $R[x;\sigma]/(x^{n+1})$. We also investigate the relationship between strongly regular rings and centrally morphic rings. For instance, we show that for a domain $R$, $R[x;\sigma]/(x^{n+1})$ is (left) centrally morphic if and only if $R$ is a division ring and $\sigma(r)=u^{-1}ru$ for some $u\in R$. Examples which delimit and illustrate our results are provided.
Keywords
Centrally morphic ring idempotent morphic ring quasi-morphic ring regular ring strongly regular ring unit-regular ring
References
- L. V. An, T. G. Nam and N. S. Tung, On quasi-morphic rings and related problems, Southeast Asian Bull. Math., 40(1) (2016), 23-34.
- V. Camillo and W. K. Nicholson, Quasi-morphic rings, J. Algebra Appl., 6(5) (2007), 789-799.
- V. Camillo, W. K. Nicholson and Z. Wang, Left quasi-morphic rings, J. Algebra Appl., 7(6) (2008), 725-733.
- H. Chen, Regular elements in quasi-morphic rings, Comm. Algebra, 39 (2011), 1356-1364.
- A. J. Diesl, T. J. Dorsey and W. W. McGovern, A characterization of certain morphic trivial extensions, J. Algebra Appl., 10(4) (2011), 623-642.
- G. Ehrlich, Units and one-sided units in regular rings, Trans. Amer. Math. Soc., 216 (1976), 81-90.
- K. R. Goodearl, Von-Neumann Regular Rings, Pitman, Boston, 1979.
- Q. Huang and J. Chen, $\pi$-morphic rings, Kyungpook Math. J., 47(3) (2007), 363-372.
- T. K. Lee and Y. Zhou, Morphic rings and unit regular rings, J. Pure Appl. Algebra, 210 (2007), 501-510.
- T. K. Lee and Y. Zhou, Regularity and morphic property of rings, J. Algebra, 322 (2009), 1072-1085.
- T. K. Lee and Y. Zhou, A theorem on unit regular rings, Canad. Math. Bull., 53(2) (2010), 321-326.
- W. K. Nicholson and E. S. Campos, Rings with the dual of the isomorphism theorem, J. Algebra, 271 (2004), 391-406.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Articles |
| Authors | |
| Publication Date | July 16, 2022 |
| Published in Issue | Year 2022 Volume: 32 Issue: 32 |