Neat submodules over integral domains (Q452829)

An exact sequence \(0\to A\to B\to C\to 0\) of abelian groups is called a neat-exact sequence if and only if (a) the sequence \[ 0\to \text{Hom}(\mathbb{Z}/p\mathbb{Z},A)\to \text{Hom}(\mathbb{Z}/p\mathbb{Z},B)\to \text{Hom}(\mathbb{Z}/p\mathbb{Z},C)\to 0 \] is exact for all primes \(p\); (b) the sequence \[ 0\to \mathbb{Z}/p\mathbb{Z}\otimes A\to \mathbb{Z}/p\mathbb{Z}\otimes B\to \mathbb{Z}/p\mathbb{Z}\otimes C\to 0 \] is exact for all primes \(p\). The definition of neatness can be extended to arbitrary commutative rings \(R\) either via (a) or (b) if we replace the groups \(\mathbb{Z}/p\mathbb{Z}\) by simple \(R\)-module \(S\). However, in general, the two definitions are not only inequivalent, but neither implies the other (see Examples 3.2 and 3.3). In the paper under review the author considers relations between the two versions in the commutative case, and characterizes the integral domains in which they coincide: these are the domains whose maximal ideals are invertible.