On serial quasi-p-injective modules. (Q1881606)

The results that a left uniserial ring or a twosided serial ring \(R\) is right self-p-injective if and only if \(J(R)=Z_r\) have been known for some years [\textit{W. K. Nicholson} and \textit{M. F. Yousif}, Bull. Aust. Math. Soc. 49, No. 3, 513--518 (1994; Zbl 0814.16014) and \textit{G. Puninski, R. Wisbauer} and \textit{M. Yousif}, Glasg. Math. J. 37, No. 3, 373--378 (1995; Zbl 0847.16005)]. Related characterizations in terms of p-injectivity are found here for right uniserial modules. For a uniserial right \(R\)-module \(M\) which is a self-generator and with endomorphism ring \(S\), necessary and sufficient conditions for \(M\) to be quasi-p-injective are found in terms of the Jacobson radical of \(S\) and other requirements on \(S\). This leads to a corollary showing that a right uniserial ring \(R\) is right self-p-injective if and only if \(J(R)=Z(R_R)\) and \(R\) is left uniserial. Thus any right uniserial right self-p-injective ring is left uniserial. An example shows that right uniserial, left self-p-injectives need not be left uniserial. It is also shown that a finitely generated quasi-projective serial right \(R\)-module \(M\) which is self-generator and has an endomorphism ring \(S\) is quasi-p-injective if and only if \(S\) is right self-projective. Application of this theorem to \(R_R\) yields the result that a right serial ring \(R\) is right self-p-injective if and only if \(J(R)=Z(R_R)\) and \(R\) is left serial. This means that every right serial right self-p-injective ring is serial. The paper concludes with the unanswered question as to whether the requirement for \(M\) to be quasi-projective may be dropped in the result above.