On orthogonal completion of reduced rings (Q789496)

It is known that the relation \(a\leq b\) if \(a\cdot b=a^ 2\) defined for reduced rings R (rings with no nonzero nilpotent elements) makes R into a partially ordered multiplicative semigroup. A subset X of R is called an orthogonal subset of R if for x,\(y\in X\), \(x\neq y\) implies \(x\cdot y=0\). A reduced ring \(\hat R\supseteq R\) is an orthogonal extension of R is every element of \(\hat R\) is the supremum of an orthogonal subset of R. An orthogonal extension \(\hat R\) of R is said to be an orthogonal completion of R if every orthogonal subset of R has a supremum in \(\hat R\). The paper's main result is a necessary and sufficient condition for a reduced ring to have an orthogonal completion.