Reflexive property on idempotents. (Q2872184)

The authors, in this paper, generalize the reflexive property for ideals to reflexive-idempotents-property (RIP) for rings. In Section 2, they derive some basic results for RIP rings. They prove that every corner ring of a RIP ring is RIP and hence deduce that if \(M_n(R)\) is RIP so is \(R\) but they have not decided the converse. They prove that a ring \(R\) is Abelian and semiperfect if and only if \(R\) is a finite direct product of local RIP rings. Using their earlier result they deduce that a right p.q.-Baer ring is RIP if and only if it is semiprime.NEWLINENEWLINE In Section 3, they study extension of RIP to polynomial and power series rings. It behaves well with both. They also prove that if \(R\) is an algebra over a commutative ring \(S\) then the Dorroh extension \(D\) of \(R\) by \(S\) is RIP if and only if \(R\) is so. If \(D_n(R)\) denotes the subring of the ring \(U_n(R)\) (of all \(n\times n\) upper triangular matrices over \(R\)) of all matrices with all the diagonal entries equal, they prove that \(D_n(R)\) is RIP if and only if \(R\) is so. They also prove that if \(M\) is a multiplicative subset of \(R\) consisting of central regular elements then if the left ring of fractions \(M^{-1}R\) of \(R\) is RIP then so is \(R\) and the converse is true under some additional hypotheses.NEWLINENEWLINE Finally in Section 4, the authors study RIP rings of minimal order. They prove that every non-Abelian RIP ring of minimal order has order 16 and is isomorphic to \(M_2(\mathbb Z_2)\). If further \(R\) has no identity then \(R\) is isomorphic to either the matrix ring \(\left(\begin{smallmatrix}\mathbb Z_2&\mathbb Z_2\\ 0&0\end{smallmatrix}\right)\) or \(\left(\begin{smallmatrix}\mathbb Z_2&0\\ \mathbb Z_2&0\end{smallmatrix}\right)\).