Some results on the comaximal ideal graph of a commutative ring

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let $R$ be a ring such that $R$ admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that the comaximal ideal graph of $R$, denoted by $\mathscr{C}(R)$ is an undirected simple graph whose vertex set is the set of all proper ideals $I$ of $R$ such that $I\not\subseteq J(R)$, where $J(R)$ is the Jacobson radical of $R$ and distinct vertices $I_{1}$, $I_{2}$ are joined by an edge in $\mathscr{C}(R)$ if and only if $I_{1} + I_{2} = R$. In Section 2 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is planar. In Section 3 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is a split graph. In Section 4 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is complemented and moreover, we determine the $S$-vertices of $\mathscr{C}(R)$

Generalized Irreducible Divisor Graphs

In 1988, I. Beck introduced the notion of a zero-divisor graph of a commutative rings with $1$. There have been several generalizations in recent years. In particular, in 2007 J. Coykendall and J. Maney developed the irreducible divisor graph. Much work has been done on generalized factorization, especially $\tau$-factorization. The goal of this paper is to synthesize the notions of $\tau$-factorization and irreducible divisor graphs in domains. We will define a $\tau$-irreducible divisor graph for non-zero non-unit elements of a domain. We show that by studying $\tau$-irreducible divisor graphs, we find equivalent characterizations of several finite $\tau$-factorization properties.Comment: 17 pages, 2 figures, to appear in Communications in Algebr

What does a group algebra of a free group know about the group?

We describe solutions to the problem of elementary classification in the class of group algebras of free groups. We will show that unlike free groups, two group algebras of free groups over infinite fields are elementarily equivalent if and only if the groups are isomorphic and the fields are equivalent in the weak second order logic. We will show that the set of all free bases of a free group $F$ is 0-definable in the group algebra $K(F)$ when $K$ is an infinite field, the set of geodesics is definable, and many geometric properties of $F$ are definable in $K(F)$. Therefore $K(F)$ knows some very important information about $F$. We will show that similar results hold for group algebras of limit groups.Comment: Published, Available for free at https://www.sciencedirect.com/science/article/pii/S0168007218300174?dgcid=STMJ_73515_AUTH_SERV_PPUB_V38 arXiv admin note: text overlap with arXiv:1509.0411