Let G be a finite graph on the vertex set [d] = {1, . . . , d} with the edges e1, . . . , en and K[t] = K[t1, . . . , td] the polynomial ring in d variables over a field K. The edge ring of G is the semigroup ringK[G] which is generated by those monomials t = titj such that e = {i, j} is an edge of G. Let K[x] = K[x1, . . . , xn] be the polynomial ring in n variables over K and define the surje...

let r be a commutative ring with identity and m be a unitary r-module. let : s(m) −! s(m) [ {;} be a function, where s(m) is the set of submodules ofm. suppose n  2 is a positive integer. a proper submodule p of m is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 r and x 2 m and a1 . . . an−1x 2p(p), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x...

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set...

The effect of alkyl substituents on the C-phenyl and/or the N-Phenyl ring of benzophenylhydroxamic acid on their molecular structure and hydrogen bonding has been investigated. The predominant configuration in CHCl3 is determined by steric and electronic effects. Substituents on the C-phenyl ring favor the cis configuration, while substituents in the N-phenyl ring favor a trans c...

Let V be a degree d, reduced hypersurface in CP, n ≥ 1, and fix a generic hyperplane, H. Denote by U the (affine) hypersurface complement, CP−V ∪H, and let U be the infinite cyclic covering of U corresponding to the kernel of the linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules Hi(U ;Q) of the hypersurface complement and show t...

Classically, an abelian group $G$ is said to be slender if every homomorphism from the countable product $\mathbb Z^{\mathbb N}\mathbb N$ factors through projection some finite Z^n$. Various authors have proposed ge

In [2], Aleksi Saarela has introduced a new approach to word equations that is based on linear-algebraic properties of polynomials encoding the equations and their solutions. In this paper we develop further this approach and take into account other algebraic properties of polynomials, namely their factorization. This, in particular, allows to improve the bound for the number of independent equ...

Hochster established the existence of a commutative noetherian ring R and a universal resolution U of the form 0 ! R ! R ! R ! 0 such that for any commutative noetherian ring S and any resolution V equal to 0! S ! S ! S g ! 0, there exists a unique ring homomorphism R ! S with V = U R S. In the present paper we assume that f = e + g and we nd a resolution of R by free P-modules, where P is a po...

Let K be a finite extension of the p-adic field Qp and let F (X,Y ) and G(X, Y ) be one-dimensional formal group laws over the ring of integers OK of K. Let φ(X) be a homomorphism from F to G which is defined over the ring of integers OCp of the completion Cp of Q p . In this paper we prove that if ker(φ) is finite then there is a discretely valued subfield L ⊂ Cp such that φ(X) is defined over...