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On (Quasi-)morphic Property of Trivial Extensions of Rings | ||
| Journal of New Researches in Mathematics | ||
| مقاله 5، دوره 8، شماره 39، خرداد و تیر 2023، صفحه 61-76 اصل مقاله (322.97 K) | ||
| نوع مقاله: research paper | ||
| شناسه دیجیتال (DOI): 10.30495/jnrm.2022.63160.2149 | ||
| نویسندگان | ||
Najmeh dehghani1؛ Mojtaba Sedaghatjoo  2
| ||
| 1Department of Mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr, Iran | ||
| 2Department of mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr, Iran | ||
| چکیده | ||
| Let R be a ring and M be an R-R bimodule. R is called left quasi-morphic if for every a∈R, there exist elements b,c∈R such that l_R (a)=Rb and l_R (c)=Ra. Besides, R is said to be left morphic whenever in the above definition b and c can be chosen the same. In this paper, we investigate conditions under which the trivial extension R⋉M of R by M is (quasi-)morphic. We present some examples showing that neither R nor M inherits the (quasi-)morphic property from R⋉M, and vice versa. So, we obtain several necessary and sufficient conditions under which R⋉M is (quasi-)morphic. For instance, we show that left quasi-morphic property of R⋉M implies that M_R is divisible. Moreover, we prove that if R⋉M is left quasi-morphic and there exists x∈M such that either r_R (x)=0 or l_R (x)=0 then (_R^ )M is cyclic. In particular, if R is commutative, then M≃R and R is also quasi-morphic. In addition, we investigate the (quasi-)morphic property of R⋉M whenever M is free as a left (right) module over R. Consequently, we prove the following theorem which is the main outcome of this paper: if R is an integral domain and (_R^ )M is free, then R⋉M is left (quasi-)morphic if and only if R is a division ring and (_R^ )M≃(_R^ )R . As an application of this theorem, the result which is proved by Lee and Zhou, and also by Van An and et al., in 2007 and 2016, respectively, is deduced. | ||
| کلیدواژهها | ||
| Annihilator؛ bimodule؛ trivial extension؛ quasi-morphic؛ morphic | ||
| مراجع | ||
|
[1]
|
J. von-Neumann, "ON REGULAR RINGS," Proc. Nat. Acad. Sci., vol. 22, pp. 707-713, 1936.
|
|
|
[2]
|
G. Ehrlich, "Unit regular rings," Portugal. Math., vol. 27, pp. 209-212, 1968.
|
|
|
[3]
|
G.Ehrlich, "UNITS AND ONE-SIDED UNITS IN REGULAR RINGS," Trans.Amer.Math.Soc., vol. 216, pp. 81-90, 1976.
|
|
|
[4]
|
W. K. Nicholson, E. S. Campos, "Rings with the dual of the isomorphism theorem," J. Algebra, vol. 271, pp. 391-406, 2004.
|
|
|
[5]
|
W. K. Nicholson and V. Camillo , "Quasi-morphic rings," J. Algebra Appl., vol. 6, no. 5, pp. 789-799, 2007.
|
|
|
[6]
|
V. Camillo, W. K. Nicholson and Z. Wang,, "Left quasi-morphic rings," J. Algebra Appl., vol. 7, pp. 725-733, 2008.
|
|
|
[7]
|
W. K. Nicholson and E. S. Campos, "Morphic Modules," Comm. Algebra, vol. 33, pp. 2629-2647, 2005.
|
|
|
[8]
|
J. Chen and Y. Zhou, "Morphic rings as trivial extensions," Glasgow Math. J., vol. 47, pp. 139-148, 2005.
|
|
|
[9]
|
J. Chen, Y. Li, Y. Zhou, "Morphic group rings," J. Pure Appl. Algebra, vol. 205, pp. 621-639, 2006.
|
|
|
[10]
|
T. J. Doresey, "Morphic and principal-ideal group rings," J. Algebra, vol. 318, pp. 393-411, 2007.
|
|
|
[11]
|
T. K. Lee and Y. Zhou, "A theorem on unit regular rings," Canad. Math. Bull., vol. 53, no. 2, pp. 321-326, 2007.
|
|
|
[12]
|
T. K. Lee and Y. Zhou, "Morphic rings and unit regular rings," J. Pure Appl. Algebra, vol. 210, no. 2, pp. 501-510, 2007.
|
|
|
[13]
|
T. K. Lee and Y. Zhou, "Regularity and morphic property of rings," J. Algebra, vol. 322, pp. 1072-1085, 2009.
|
|
|
[14]
|
L. V. An, T. G. Nam and N. S. Tung, "On quasi-mrphic rings and Related problems," Southest Asian Bull. Math., vol. 40, pp. 23-34, 2016.
|
|
|
[15]
|
N. Dehghani, "On (quasi-)morphic property of skew polynomial rings,"Int. Electron. J. Algebra, Published Online: 12 April 2022
|
|
|
[16]
|
W. K. Nicholosn, J. K. Park and M. F. Yousif , " Extensions of Simple-injective Rings," Comm. Algebra,, vol. 28, no. 10, pp. 4665-4675, 2008.
|
|
|
[17]
|
C. Faith , "Self-injective Rings," Prue. Amer. Math. Sec., vol. 77, no. 2, pp. 157-164, 1979.
|
|
|
[18]
|
W. Yunxia, C. Jianlong, " On trivial extensions of rings". Nanjin Daxue Xuebao Shuxue Bannian Kan., vol. 22, no. 2, pp. 234-243, 2005.
|
|
|
[19]
|
K. R. Goodearl, Von-Neumann Regular Rings, Pitman, 1979.
|
|
آمار
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