On (Quasi-)morphic Property of Trivial Extensions of Rings

Subject Areas : Statistics

Najmeh  dehghani ¹ (Department of Mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr, Iran)
Mojtaba  Sedaghatjoo ² (Department of mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr, Iran)

Received: 2021-09-16 Accepted : 2022-06-01 Published : 2022-12-22

Keywords: دو مدول, توسیع بدیهی, پوچ‌ساز, شبه-مورفیک, مورفیک,

Abstract :

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.

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