Additively cancellative congruences on hemirings (Q1281314)

A hemiring \((S,+,\cdot)\) is an additively commutative semiring with absorbing zero. It is well known that a hemiring \(S\) is embeddable into a ring iff \((S,+)\) is cancellative. In this case, there exists a smallest ring of this kind, the ring \(D(S)\) of differences of \(S\). A congruence \(\varrho\) on \(S\) is called additively cancellative (AC) if \((a+x,b+x)\in\varrho\) implies \((a,b)\in\varrho\) for all \(a,b,x\in S\). Note that there is always a smallest AC congruence on \(S\), namely \(\overline\Delta=\{(x,y)\in S\times S\mid x+r=y+r\) for some \(r\in S\}\). Hence the quotient hemiring \(S/\overline\Delta\) is additively cancellative and the difference ring \(D(S/\overline\Delta)\) exists. An ideal \(I\) of \(S\) is called an \(h\)-ideal if \(x+a+z=b+z\) for some \(x,z\in S\) and \(a,b\in I\) implies \(x\in I\) . The main result is: There is a one-to-one correspondence between the set of all \(h\)-ideals of \(S\) and all AC congruences on \(S\) iff \(D((\overline x)\cap S/\overline\Delta)=(\overline x)\) for all \(\overline x\in D(S/\overline\Delta)\), where \((\overline x)\) denotes the principal ring ideal generated by \(\overline x\) in \(D(S/\overline\Delta)\).