Reflexive rings and their extensions. (Q2843880)

Let \(R\) be a ring. Following Mason, a right ideal \(I\) of \(R\) is said to be reflexive if \(xRy\subseteq I\) implies \(yRx\subseteq I\) for \(x,y\in R\). In the literature, \(R\) is called reflexive if the zero ideal is reflexive. Following the authors, a right ideal \(I\) of \(R\) is called completely reflexive if \(xy\in I\) implies \(yx\in I\) for \(x,y\in R\); and \(R\) is called completely reflexive if \(xy=0\) implies \(yx=0\) for \(x,y\in R\). But ``reversible'' has been used for what is called ``completely reflexive''.NEWLINENEWLINE The authors investigate the structure of reflexive rings related to ordinary ring extensions which have roles in ring theory. The authors first prove the following: (1) Some kind of subring of \(T_3(R)\) over a reduced ring \(R\) can be reflexive, comparing with the fact that \(U_3(A)\) is not reflexive over any ring \(A\), where \(T_3(R)\) is the \(3\) by \(3\) upper triangular matrix ring over a ring \(R\).NEWLINENEWLINE (2) If \(R\) is a reduced ring, then \(R[x]/(x^n)\) is a reflexive ring, where \(R[x]\) is the polynomial ring with an indeterminate \(x\) over \(R\) and \((x^n)\) is the ideal of \(R[x]\) generated by \(x^n\).NEWLINENEWLINE (3) Let \(R\) be a ring and \(\Delta\) be a multiplicative monoid of \(R\) consisting of central regular elements. Then \(R[x]\) is reflexive if and only if so is \(\Delta^{-1}R[x]\).NEWLINENEWLINE (4) Let \(R\) be a right Ore ring with \(Q\) the classical right quotient ring of \(R\). Then if \(R\) is reflexive then so is \(Q\).NEWLINENEWLINE The authors next introduce a new concept as a generalization of reflexivity. A ring \(R\) is called weakly reflexive if \(aRb=0\) implies \(bRa\subseteq\mathrm{nil}(R)\) for \(a,b\in R\), where \(\mathrm{nil}(R)\) means the set of all nilpotent elements in \(R\).NEWLINENEWLINE They prove the following: (1) If \(R\) is a weakly reflexive ring then \(T_n(R)\) is weakly reflexive for all \(n\geq 1\).NEWLINENEWLINE (2) Let \(R\) be a ring. \(R[x]\) is weakly reflexive if and only if \(R[x;x^{-1}]\) is weakly reflexive.NEWLINENEWLINE (3) If a ring \(R\) is semicommutative then \(R[x]\) is weakly reflexive, where a semicommutative ring \(R\) is usually defined by that \(ab=0\) implies \(aRb=0\) for \(a,b\in R\).