For a commutative semigroup S with 0, the zero-divisor graph of S denoted by &Gamma(S) is the graph whose vertices are nonzero zero-divisor of S, and two vertices x, y are adjacent in case xy = 0 in S. In this paper we study median and center of this graph. Also we show that if Ass(S) has more than two elements, then the girth of &Gamma(S) is three.

Abstract We study the dimension of loci special line bundles on stable curves and for a fixed semistable multidegree. In case total degree $$d = g - 1$$ d = g - 1 , we characterize when effective locus gives Theta divis...

Contents 1. Introduction 2 2. Prym varieties for covers of curves 3 3. Galois covers 8 4. Degree two covers 10 5. Covers of degree three 13 6. Covers of degree four 15 6.1. The cyclic case 15 6.2. The Klein case 17 7. The dihedral case 22 7.1. The bigonal construction 34 8. The alternating case 37 8.1. The trigonal construction for the case A 4 43 9. The symmetric case 44 9.1. The classical cas...

For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...

Let K be an octic number field generated by a complex root $$\theta$$ of monic irreducible trinomial $$F(x)= x^{8}+ax+b \in \mathbb{Z}[x]$$ , where and b are two non-zero rational integers. $$i(K)$$ the index K. We show that i(K) is either 1 or power 2. Further, assuming $$(a,b) \notin (32+64\mathbb{Z})\times {(16+64\mathbb{Z}) } $$ (64\mathbb{Z})\times(112+128\mathbb{Z})$$ we give necessary su...