Arithmetical rank of the edge ideals of some n-cyclic graphs with a common edge

In this paper, we present some lower bounds and upper bounds on the arithmetical rank of the edge ideals of some n-cyclic graphs with a common edge. For some special n-cyclic graphs with a common edge, we prove that the arithmetical rank equals the projective dimension of the corresponding quotient ring. | Turkish Journal of Mathematics Research Article Turk J Math (2015) 39: 112 – 123 ¨ ITAK ˙ c TUB ⃝ doi: Arithmetical rank of the edge ideals of some n-cyclic graphs with a common edge Guangjun ZHU∗, Feng SHI, Yan GU School of Mathematical Sciences, Soochow University, Suzhou, . China Received: • Accepted: • Published Online: • Printed: Abstract: In this paper, we present some lower bounds and upper bounds on the arithmetical rank of the edge ideals of some n -cyclic graphs with a common edge. For some special n -cyclic graphs with a common edge, we prove that the arithmetical rank equals the projective dimension of the corresponding quotient ring. Key words: Arithmetical rank, edge ideal, projective dimension, set-theoretic complete intersection 1. Introduction Let R be a Noetherian commutative ring with identity and I a proper ideal of R . The arithmetical rank (ara) of I is defined as the minimal number s of elements a1 , . . . , as of R such that the ideal (a1 , . . . , as ) has the same radical as I . In this case we will say that a1 , . . . , as generate I up to radical. In general ht (I) ≤ ara (I). If equality holds, I is called a set-theoretic complete intersection. We consider the case where R is a polynomial ring over any field K and I is the edge ideal of a graph whose vertices are the indeterminates. The set of its generators is formed by the products of the pairs of indeterminates that form the edges of the graph. Thus, I is generated by square-free quadratic monomials and is therefore a radical ideal. The problem of the arithmetical rank of edge ideals or monomial ideals has been intensively studied by many authors over the past 3 decades (see [1, 2, 3, 5, 8, 10]). According to a well-known result by Lyubeznik [9], if I is a square-free monomial ideal, the projective dimension of the quotient ring R/I , denoted pd R (R/I), provides a lower .

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