Coneat submodules and coneat-flat modules. (Q2929764)

In a paper by \textit{L. Fuchs} [Period. Math. Hung. 64, No. 2, 131-143 (2012; Zbl 1289.13011)], a submodule \(N\) of a right \(R\)-module \(M\) was called coneat in \(M\) if \(\Hom(M,S)\to\Hom(N,S)\to 0\) is an epimorphism for each simple right \(R\)-module \(S\). This is the dual of the earlier concept a neat submodule. Coneat submodules are investigated further in the paper under review, particularly the closure properties of the class of coneat submodules. Several characterizations of coneatness are found for submodules of modules over particular types of rings. For example, if \(R\) is a commutative ring, then a short exact sequence is shown to be coneat exact if and only if the induced sequence of character modules is neat exact.NEWLINENEWLINE In analogy to the concept of flatness, a right \(R\)-module \(M\) is defined to be coneat-flat if for any epimorphism \(\psi\colon Y\to M\), the sequence \(0\to\ker\psi\to Y\to M\to 0\) is coneat exact. The coneat-flat modules \(M\) are characterized as those modules \(M\) such that \(\mathrm{Ext}_R^1(M,S)=0\) for every simple \(R\)-module \(S\). For the case where \(R\) is commutative, an \(R\)-module \(M\) is shown to be coneat-flat if and only if the character module \(M^+\) is \(m\)-injective. The rings for which each right \(R\)-module is coneat-flat, are shown to be exactly the right \(V\)-rings.NEWLINENEWLINE An example shows that not all flat modules are coneat-flat. Neither is each coneat-flat module a flat module. It is shown that, over specific types of rings, flat modules are indeed coneat-flat or the other way round. Over a commutative ring \(R\) an equivalence is found for each coneat-flat module to be flat. The relationship between coneat-flatness of \(M\), \(M^+\) and \(M^{++}\) is also studied for modules over commutative \(N\)-rings.NEWLINENEWLINE The final section studies the question as to when a coneat-flat module is projective and it is found that for finitely presented modules over a commutative ring, coneat-flatness and projectivity coincide.