Zero-divisor graphs of nilpotent-free semigroups
- First Online:
- 222 Downloads
We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an Armendariz map between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs.
KeywordsZero-divisor graph Armendariz map Graph invariants Annihilating-ideal graph Comaximal graph Nilpotent-free semigroup
- 1.Aalipour, G., Akbari, S., Nikandish, R., Nikmehr, M., Shaveisi, F.: On the coloring of the annihilating-ideal graph of a commutative ring. Discrete Math. (2011). doi:10.1016/j.disc.2011.10.020
- 14.Lu, D., Wu, T., Ye, M., Yu, H.: On graphs related to the co-maximal ideals of a commutative ring (2011). arXiv:1106.0072v1