2-good rings and their extensions. (Q2873008)

In this paper, the authors study 2-good rings, introduced and studied by P. Vamos, and study their extensions. After introduction in Section 1, in Section 2, they list results on 2-good rings in this new version (earlier authors studied these rings without giving this name). Here the authors prove that every exchange ring with primitive factors Artinian with 2 as a unit is 2-good. This generalizes the result of Fisher-Snider and Ehrlich. They also prove that a ring \(R\) is 2-good if and only if all its Pierce stalks are 2-good. Their Corollary 2.16 is not meaningful since \(eR\) need not have identity element and we can not talk of 2-goodness of \(eR\) unless \(e\) is central in which case there is nothing to prove.NEWLINENEWLINE In Section 3, they study extension of 2-goodness to polynomial and power series rings. They prove that for a (semi)commutative ring \(R\), the polynomial ring \(R[X]\) is never 2-good. However, \(R[[X]]\) is 2-good if and only if \(R\) is so. They prove that 2-goodness carries over to full matrix rings as well as to upper triangular matrix rings. If \(M\) is a \((R,R)\)-bimodule the authors prove that the trivial extension ring \(T(R,M)\) is 2-good if and only if \(R\) is so. They also prove that if \(R\) is an algebra over a commutative ring \(S\) then the Dorroh extension \(D(R,S)\) of \(R\) by \(S\) is 2-good if \(S\) is 2-good and \(R\) is a right quasi-regular ideal of \(D(R,S)\). They conclude the paper with the result that the classical right quotient ring \(Q\) of a right Ore ring \(R\) is 2-good if \(R\) is so but the converse is not true.NEWLINENEWLINE In the beginning of the paper the authors say that \(N(R)\) denotes the nil radical of a ring \(R\). Here authors may mean \(N(R)\) is the set of nilpotent elements of \(R\). In the noncommutative case there is no nil radical of a ring but lower nil radical \(\text{Nil}_*(R)\) and upper nil radical \(\text{Nil}^*(R)\) of a ring are defined.