On a class of modules (Q2714296)

For \(R\) a ring with identity and \(M\) a unital right \(R\)-module, \(Z^*(M)=\{m\in M:mR\) small\}, the author studies the property (T): ``for every right \(R\)-module \(M\) with \(Z^*(M)=\text{Rad }M\), \(M\) is injective''.NEWLINENEWLINENEWLINEThe main result of the paper is that for a prime PI-ring \(R\) the following are equivalent: (i) \(R\) satisfies (T); (ii) for every left \(R\)-module with \(Z^*(M)=\text{Rad }M\), \(M\) is injective; (iii) \(R\) is a hereditary Noetherian ring.NEWLINENEWLINENEWLINEAs a consequence the authors proves that over a right Noetherian ring \(R\), if \(R\) satisfies (T) then every right \(R\)-module is the direct sum of an injective module and a max-module.