Edge ideals associated to unit graphs

Time: 09:30 to  10:30 Ngày 29/03/2018

Venue/Location: B4-705, VIASM

Speaker: M. R. Pournaki

Content:

The ring $\mathbb{Z}_2\times\mathbb{Z}_2$, having only one unit, cannot be generated by its units. It turns out, in the general theory of rings, that this is essentially the only example. In this talk, we give an elementary proof of ``A finite commutative ring with nonzero identity is generated by its units if and only if it cannot have $\mathbb{Z}_2\times\mathbb{Z}_2$ as a quotient." The proof uses graph theory and unit graphs will be arisen. At the end, we discuss some properties of unit graphs and propose some questions regarding the edge ideals associated to these graphs.