Some remarks on non-commutative principal ideal rings
نویسندگان
چکیده
منابع مشابه
The principal ideal subgraph of the annihilating-ideal graph of commutative rings
Let $R$ be a commutative ring with identity and $mathbb{A}(R)$ be the set of ideals of $R$ with non-zero annihilators. In this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of $R$, denoted by $mathbb{AG}_P(R)$. It is a (undirected) graph with vertices $mathbb{A}_P(R)=mathbb{A}(R)cap mathbb{P}(R)setminus {(0)}$, where $mathbb{P}(R)$ is...
متن کاملThe Intersection Graph of Finite Commutative Principal Ideal Rings
In this article we consider the intersection graph G(R) of nontrivial proper ideals of a finite commutative principal ideal ring R with unity 1. Two distinct ideals are adjacent if they have non-trivial intersection. We characterize when the intersection graph is complete, bipartite, planar, Eulerian or Hamiltonian. We also find a formula to calculate the number of ideals in each ring and the d...
متن کاملSome Properties of the Intersection Graph for Finite Commutative Principal Ideal Rings
Let R be a commutative finite principal ideal ring with unity, and letG(R) be the simple graph consisting of nontrivial proper ideals of R as vertices such that two vertices I and J are adjacent if they have nonzero intersection. In this paper we continue the work done by Abu Osba. We calculate the radius, eccentricity, domination number, independence number, geodetic number, and the hull numbe...
متن کاملMATH 250B: ALGEBRA SEMISIMPLICITY 1. Remarks on non-commutative rings
We begin with some reminders and remarks on non-commutative rings. First, recall that if R is a not necessarily commutative ring, an element x ∈ R is invertible if it has both a left (multiplicative) inverse and a right (multiplicative) inverse; it then follows that the two are necessarily equal. Note however, that unlike the example of n× n matrices over a field, in a general noncommutative ri...
متن کاملExact annihilating-ideal graph of commutative rings
The rings considered in this article are commutative rings with identity $1neq 0$. The aim of this article is to define and study the exact annihilating-ideal graph of commutative rings. We discuss the interplay between the ring-theoretic properties of a ring and graph-theoretic properties of exact annihilating-ideal graph of the ring.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Comptes Rendus Mathematique
سال: 2013
ISSN: 1631-073X
DOI: 10.1016/j.crma.2013.01.006