On the diameter of the compressed zero-divisor graph

Let R be acommutativeringwithnonzeroidentity. Therelationon R givenby a similar to b if andonlyif annR(a) = annR(b) is anequivalencerelation. Thecompressedzero- divisorgraph WE(R) of R is the(undirected) graphwithvertices the equivalenceclassesinducedby similar to other than [0](R) and [1](R), anddistinct vertices [a](R) and [b](R) areadjacentifandonlyif ab = 0. Thedistancebetween vertices [a](R) and [b](R) (not necessarilydistinctfrom a) isthelengthofthe shortestpathconnectingthem, andthediameterofthegraph, diam(inverted right prependiculr (E)(R)), is thesupofthesedistances. Inthispaper, wecontinuestudyofthediameter of thecompressedzero-divisorgraph inverted right prependiculr(E)(R). Acompletecharacterizationfor the possiblediametersof inverted right prependiculr(E)(R) is givenexclusivelyintermsoftheideals of R. Alsowegiveacompletecharacterizationforthepossiblediameters of inverted right prependiculr(E)(R[x]) in terms of the diameters of inverted right prependiculr(E)(R). Forareducedring R with nonzerozero-divisors, itisshownthat1 <= diam(inverted right prependiculr (E)(R)) = diam(inverted right prependiculr(E)(R[x])) <= diam(inverted right prependiculr (E)(R[[x]])) <= 3.