Bézout domains and skew polynomials (Q2780500)

Let \(A\) be an associative ring with non-zero identity and let \(\varphi\) be an injective endomorphism of the ring \(A\). By \(A_r[x,\varphi]\) is denoted the right ring of skew polynomials consisting of finite formal sums \(\sum_ix^ia_i\) in a variable \(x\) with coefficients \(a_i\in A\), where the multiplication is defined by the rule \(ax^i=x^i\varphi^i(a)\). A module is called a Bézout module if each of its finitely generated submodules is cyclic.NEWLINENEWLINENEWLINEThe author shows that the following conditions are equivalent: (1) \(A_r[x,\varphi]\) is a right Bézout domain; (2) \(A_r[x,\varphi]\) is a right Bézout ring and \(A\) a domain; (3) \(A_r[x,\varphi]/x^2A_r[x,\varphi]\) is a right Bézout ring and \(A\) a domain; (4) \(A\) is a right Bézout domain and \(\varphi(a)A=A\) for every non-zero \(a\in A\).