Strong primeness in matrix rings.
A ring \(R\) is right strongly prime if for every nonzero \(a\in R\), there exists a nonempty finite subset \(S\) of \(R\) (called a right insulator for \(a\)) such that the set \(aS\) has a trivial right annihilator. A ring \(R\) is right strongly prime of bound \(n\) if there exists a positive integer \(n\) such that every nonzero element in \(R\) has a right insulator of size \(n\) and no smaller such \(n\) exists. Left strongly prime rings and left insulators are defined in a dual manner. A ring \(R\) is uniformly strongly prime if \(R\) contains a finite subset \(S\) (called a uniform insulator for \(R\)) that is a right insulator for every nonzero \(a\in R\). A ring \(R\) is uniformly strongly prime of bound \(n\) if \(n\) is the smallest positive integer for which \(R\) has a uniform insulator of size \(n\). The bound of uniform strong primeness of the ring \(\mathbb M_n(R)\) of all \(n\) by \(n\) matrices over a unitary ring \(R\) is denoted by \(m_n(R)\). In this paper the authors develop a method for the calculation of \(m_n(R)\) that involves reduction to a system of bilinear equations over \(R\) and, using this method, they obtain some new results concerning bounds of uniform strong primeness in matrix rings. In particular, the authors investigate the bound \(m_2(D)\) for a division ring \(D\) and show that \(m_2(\mathbb H)=3\), where \(\mathbb H\) denotes the ring of quaternions. They define a division pseudoalgebra over a division ring \(D\) to be a \(D\)-bimodule \(_DM_D\) with a multiplication \(\odot\colon M^2\to M\) which is left linear in its first argument, right linear in its second and such that for \(y\in M\setminus\{0\}\), \(x\in M\), each of the equations \(y\odot w=x\) and \(w\odot y=x\) has exactly one solution \(w\in M\). They show that for every natural number \(n\) and every division ring \(D\), there exists a division pseudoalgebra over \(D\) which is isomorphic to \(D^n\) as a \(D\)-module precisely when \(m_n(D)=n\). The authors prove that for a unitary ring \(R\) and natural numbers \(n\) and \(n'\) we have \(m_{nn'}(R)\leq (2n-1)m_{n'}(R)\) and for formally real fields \(F\), we have \(m_n(F)\leq 2n-2\) for integers \(n>1\) and \(m_n(F)\leq 2n-4\) for even \(n>2\). Consequently, they obtain that \(m_{2^k+1}(\mathbb R)=m_{2^k+2}(\mathbb R)=2^{k+1}\) and also give bounds on \(m_n(\mathbb R)\) for other \(n\), where \(\mathbb R\) denotes the set of all real numbers.
scientific article; zbMATH DE number 6350929