Warfield rings (Q1892816)

This paper considers a question posed by \textit{R. B. Warfield jun.} [J. Algebra 37, 187-222 (1975; Zbl 0319.16025), p. 197]\ as to which rings \(R\) have the property that every finitely presented right \(R\)-module is a direct summand of a direct sum of cyclic finitely presented modules; the author calls such rings right \(W1\)-rings. \textit{R. B. Warfield jun.} [Proc. Am. Math. Soc. 25, 167-172 (1970; Zbl 0204.059)]\ proved that the commutative \(W1\)-rings are exactly the Prüfer rings. A solution is given using model-theoretic techniques, working within the general framework of \(S\)-purity, where \(S\) is a class of finitely presented modules. Several characterizations are given including an elimination of quantifiers result, giving a simplified form for positive primitive formulas; the author uses this to prove that many classes of local \(W1\)-rings are serial, including the Noetherian local \(W1\)-rings and the left perfect local \(W1\)-rings. It is pointed out that it remains open whether every local \(W1\)-ring is a serial ring and also whether every right \(W1\)-ring is a left \(W1\)-ring.