FP-injective quotient rings and elementary divisor rings (Q2785938)

For any ring property \({\mathcal P}\), a ring \(R\) is termed as fractionally \({\mathcal P}\) iff the classical quotient ring \(Q(R/I)\) has \({\mathcal P}\) for all ideals \(I\) of \(R\). \(R\) is said to be locally \({\mathcal P}\) if the localization \(R_M\) of \(R\) has \({\mathcal P}\) for every maximal ideal \(M\) of \(R\). An \(R\)-module \(A\) is \(FP\)-injective if \(\text{Ext}^1_R (F,A)=0\) for every finitely presented \(R\)-module \(F\). A commutative ring \(R\) is a valuation ring if the set of its ideals is linearly ordered under inclusion and \(R\) is called an arithmetical ring if it is locally a valuation ring. -- The authors study (commutative) fractionally self \(FP\)-injective rings which they denote by \(FSFPI\) rings. They show that every \(FSFPI\) commutative ring is arithmetical. It is known that arithmetical rings need not be \(FSFPI\). A construction is provided by the authors which yields examples of arithmetical rings with \(Q(R)= R\), which are not self \(FP\)-injective.NEWLINENEWLINENEWLINEIt is proved that a commutative ring \(R\) is a \(FSFPI\) ring iff \(R\) is an arithmetical ring provided \(R\) is one of the following types: (i) a zero-dimensional ring, (ii) a one-dimensional integral domain, (iii) a fractionally semilocal ring, or (iv) a semilocal ring. -- As a consequence, one obtains that a noetherian commutative ring is \(FSFPI\) if and only if it is a finite direct product of Dedekind domains and local artinian principal ideal rings. The authors also prove that any elementary divisor ring \(R\) satisfying one of the conditions (i)-(iv) listed above, is \(FSFPI\). For the converse, it is shown that any semilocal \(FSFPI\) ring is an elementary divisor ring. It is proved that an arithmetical or coherent self \(FP\)-injective ring \(R\) is always locally self \(FP\)-injective and the converse holds in case \(R\) is semilocal arithmetical or \(R\) is coherent.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].