Generalizations of injective modules. (Q2920234)

A right \(R\)-module \(M\) is said to be \(n\)-injective (resp. F-injective; resp. C-injective) if for every \(n\)-generated (resp. finitely generated; resp. countably generated) right ideal \(A\) of \(R\), every homomorphism \(\varphi\colon A\to M\) lifts to \(R\). A right \(R\)-module \(X\) is called \(nP\)-injective (resp. FP-injective) if for every free (resp. finitely generated free) \(R\)-module \(F\) and \(n\)-generated (resp. finitely generated) submodule \(G\) of \(F\), every homomorphism \(\psi\colon G\to X\) can be lifted to \(F\). Relationships between these injectivities are first pointed out and then properties of the different types of injectivity, such as closure under direct summands, direct sums and direct products are investigated. For a semiprime right Goldie ring, every torsionfree 1-injective ring is shown to be injective.NEWLINENEWLINE A ring \(R\) is called right semihereditary (resp. right \(n\)-semihereditary) if every finitely generated (resp. \(n\)-generated) right ideal is projective. For a right \(n\)-semihereditary ring \(R\), the concepts of \(n\)-injectivity and \(nP\)-injectivity are shown to be equivalent. 1-injectivity is studied in depth. Over a commutative ring \(R\), every 1-injective simple \(R\)-module is shown to be injective.NEWLINENEWLINE Let \(e\) be an idempotent in a ring \(R\), such that \(R=ReR\). If \(M\) is a right \(R\)-module, then \(Me\) is a unitary right module over the subring \(S=eRe\) of \(R\). It is shown that if \(M\) is an \(n\)-injective (resp. \(nP\)-injective; resp. C-injective; resp. F-injective; resp. FP-injective) right \(R\)-module (\(n\) some positive integer where applicable), then \(Me\) is an \(n\)-injective (resp. \(nP\)-injective; resp. C-injective; resp. F-injective; resp. FP-injective) right \(S\)-module.NEWLINENEWLINE The question as to whether, for any positive integer \(n\), there exists a ring \(R\) such that \(R\) is right \(n\)-injective, but not right \((n+1)\)-injective, is investigated next. For a ring \(R\) that is right \(n\)-semihereditary but not right \((n+1)\)-semihereditary, for some positive integer \(n\), an \(R\)-module is constructed that is \(nP\)-injective but not \((n+1)\)-injective. This means that it is sufficient to construct rings \(R\) which are right \(n\)-semihereditary but not right \((n+1)\)-semihereditary in order to answer the above question. To this end, it is shown that for every field and positive integer \(n\), an \(F\)-algebra \(A\) exists which is a right \(n\)-semihereditary domain, but is not right \((n+1)\)-semihereditary. Using this algebra \(A\), many other rings with the same property are constructed.