Endpiece: The ideal

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...

The Dual of a Strongly Prime Ideal

Let R be a commutative integral domain with quotient field K and let P be a nonzero strongly prime ideal of R. We give several characterizations of such ideals. It is shown that (P : P) is a valuation domain with the unique maximal ideal P. We also study when P^{&minus1} is a ring. In fact, it is proved that P^{&minus1} = (P : P) if and only if P is not invertible. Furthermore, if P is invertib...

Cross-efficiency evaluation by the use of ideal and anti-ideal virtual DMUs’ assessment in DEA

To rank Decision Making Units (DMU) in Data Envelopment Analysis (DEA), the peer-evaluation based cross-efficiency method is generally used. Indeed, in this method, each DMU is evaluated from the view point of other DMUs. In this article, a method is suggested which instead examines each DMU just by using the weights resulting from the evaluation of ideal and anti-ideal virtual DMUs and thus ex...

Ranking Decision Making Units with the ideal and anti-ideal points

This paper introduces two virtual Decision Making Units (DMUs) called ideal point and anti-ideal point, Then calculates distances of each DMU to the ideal and anti-ideal point. The two distinctive distances are combined to form a comprehensive index called the relative closeness (RC) just like the TOPSIS approach. The RC index is used as an overall ranking for all the DMUs. Then, this method co...

The Multiplier Ideal Is a Universal Test Ideal