Linear recurring sequences over noncommutative rings. (Q2885401)

Let \(R\) be a ring and let \(M\) be an \(R\)-module. Then the set \(S(M)\) of sequences with entries in \(M\) is clearly an \(R\)-module. In fact, it is a module over the polynomial ring \(R[x]\), where \(x\) acts by \((xu)(n)=u(n+1)\). Let \(LRS_R(M)\) be the set of all \textit{linear recurring sequences}; that is, \(LRS_R(M)\) consists of all sequences \(u\) over \(M\) such that there is a monic \(p(x)\in R[x]\) with \(p(x)u=0\). If \(R\) is commutative, then clearly \(LRS_R(M)\) is an \(R\)-submodule of \(S(M)\).NEWLINENEWLINE If \(R\) is not commutative, then \(LRS_R(M)\) may fail to be a submodule of \(S(M)\). In fact, if \(\alpha,\beta\in R\) and if \(u(n)=\alpha^n\) and \(v(n)\in\beta^n\), then \(u,v\in LRS_R(R)\), and the authors show that if \(u+v\in LRS_R(R)\), then \(R\alpha\cap R\beta\neq (0)\). This shows, in particular, that if \(R\) is the free algebra over a commutative ring \(k\) in two variables \(x,y\), then \(LRS_R(R)\) is not closed under addition (or scalar multiplication). This condition on left ideals is the same as the Ore condition of noncommutative localization theory when \(R\) is a domain. Recall that if \(S\) is the set of regular elements of \(R\), then \(R\) satisfies the left Ore condition (i.e., is a left Ore ring) if for each \(r\in R\) and \(s\in S\), we have \(Rs\cap Sr\neq\emptyset\). The ring \(R\) has a (left) ``classical'' ring of quotients precisely when it satisfies the left Ore condition.NEWLINENEWLINE In this paper the authors consider for which rings \(R\) is \(LRS_R(M)\) a submodule of \(S(M)\). They prove that if \(M_n(k)\) is the ring of \(n\times n\) matrices over a commutative ring \(k\), or if \(R\) is a division ring, then \(LRS_R(R)\) is an \(R\)-module. Their division ring argument shows that if \(R\) is a left Ore domain, then \(LRS_R(R)\) is a submodule of \(S(M)\). They mention, as a corollary to their results, that if \(A\) is a subring of \(M_n(k)\) and is a domain, then \(A\) is a left Ore domain. This result also follows from the theory of polynomial identities, as it is known that any prime PI ring is left and right Ore.