Zero-divisor placement, a condition of Camillo, and the McCoy property (Q2220182)

Let \(R\) be a (not necessarily commutative) ring and form \(R[x]\) the ring of polynomials over \(R\), where \(x\) commutes with the elements of \(R\). The ring \(R\) is a McCoy ring whenever for every \(f\), \(g\in R[x]\) with \(g\ne 0\) but \(fg=0\), then \(fr=0\) for some \(0\ne r\in R\). In other words, a nontrivial right annihilator in \(R[x]\) must contain nonzero elements of \(R\). This notion was motivated by the well known result of McCoy for commutative rings. In fact the above defines the right McCoy rings while the left ones are defined similarly. The paper under review studies generalizations of the notion of a McCoy ring. Several of them are well known; the authors recall these in order to justify their interest in the topic. A ring \(R\) is reversible if for every \(a\), \(b\in R\), the equality \(ab=0\) implies \(ba=0\). This condition implies the McCoy property for \(R\). Another example is represented by the right duo rings: these are rings where every right ideal is also a left ideal. (A duo ring is left and right duo ring.) It is known that every right duo ring is right McCoy but it is an open question whether a right duo ring must be left McCoy. A ring \(R\) is semicommutative if for every \(a\), \(b\in R\) with \(ab=0\) one has \(aRb=0\). This condition does not imply that \(R\) is McCoy but if \(R[x]\) is semicommutative then \(R\) is McCoy. The authors deal with the so called outer McCoy condition. It is similar to the original McCoy condition but the scalar \(r\) is written on the left of \(f\) instead of the right-hand side of \(f\). First, they relate this condition to the ones listed above. They prove that such rings are reversible, hence the commutative and the reduced rings are outer McCoy. They prove that every outer McCoy ring is McCoy. Apart from these positive results, the authors construct a ring which is duo and Armendariz, right outer McCoy but not left outer McCoy. Furthermore, they give an example of a semicommutative, left McCoy, and right outer McCoy ring which is neither right McCoy nor left outer McCoy. It should be noted that both examples are rather nontrivial and involved. Finally, the authors study the so-called Camillo rings. These are rings such that for every nonzero \(f\), \(g\in R[x]\) with \(fg=0\) there exists \(0\ne r\in R\) with either \(rf=0\) or \(rg=0\). (This is the definition of a left Camillo ring, the right ones are defined similarly.) These are quite natural: if in a ring the prime radical coincides with the set of all nilpotent elements then this ring is Camillo. The authors prove several interesting properties of Camillo rings. They also give an example of a ring that is left McCoy and left outer McCoy, but not right linearly Camillo. It turns out that no matrix ring can be a Camillo ring (but matrices over division rings are linearly Camillo). There are several other interesting positive results and clever examples which could motivate (or discard) attempts to generalize several of the notions considered in the paper.