A note on dual Goldie dimension (Q1181428)

This paper answers three questions about the dual Goldie dimension of an \(R\)-module \(M\), denoted by \(\hbox{corank}(_ RM)\), that were raised by \textit{B. Sarath} and \textit{K. Varadarajan} [Commun. Algebra 7, 1885-1899 (1979; Zbl 0487.16020)] and \textit{K. Varadarajan} [ibid. 10, 2223-2231 (1982; Zbl 0495.16011)]. Let \(R\) be a ring with identity element. If \(_ RP\) is a quasi-projective module with \(J(P)\) small in \(P\) and if \(S=\hbox{End}(_ RP)\), then \(\hbox{corank}(_ RP)=\hbox{corank}(_ SS)\). Further, \(\hbox{corank}(_ RR)=\hbox{corank}(_{R[[x]]}R[[x]])\). For a ring \(T\) without identity, \(\hbox{corank}(_ TT)\) and \(\hbox{corank}(T_ T)\) can be any two natural numbers or infinity.