### A Change of Ring Theorem with Applications to Poincaré Series and Intersection Multiplicity.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Butler groups formed by factoring a completely decomposable group by a rank one group have been studied extensively. We call such groups, bracket groups. We study bracket modules over integral domains. In particular, we are interested in when any bracket $R$-module is $R$ tensor a bracket group.

Let $K$ be a field, $A=K[{X}_{1},\cdots ,{X}_{n}]$ and $\mathbb{M}$ the set of monomials of $A$. It is well known that the set of monomial ideals of $A$ is in a bijective correspondence with the set of all subsemiflows of the $\mathbb{M}$-semiflow $\mathbb{M}$. We generalize this to the case of term ideals of $A=R[{X}_{1},\cdots ,{X}_{n}]$, where $R$ is a commutative Noetherian ring. A term ideal of $A$ is an ideal of $A$ generated by a family of terms $c{X}_{1}^{{\mu}_{1}}\cdots {X}_{n}^{{\mu}_{n}}$, where $c\in R$ and ${\mu}_{1},\cdots ,{\mu}_{n}$ are integers $\ge 0$.

In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring $R$, denoted by $G\left(R\right)$, is a graph with all non-small proper ideals of $R$ as vertices and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J$ is not small in $R$. In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter,...