Genus theory belongs to algebraic number theory and, in very broad terms, deals with the part of the ideal class group of a number field that is ‘easy to compute’. Historically, the importance of genus theory stems from the fact that it was the essential algebraic ingredient in the derivation of the classical reciprocity laws – from Gauß’s second proof over Kummer’s contributions up to Takagi’s...

In this paper, after reviewing some results in minimal space, some new results in this setting are given. We prove a generalized form of the Fan-KKM typetheorem in minimal vector spaces. As some applications, the open type of matching theorem and generalized form of the classical KKM theorem in minimal vector spaces are given.

Last week, Ari taught you about one kind of “simple” (in the nontechnical sense) ring, specifically semisimple rings. These have the property that every module splits as a direct sum of simple modules (in the technical sense). This week, we’ll look at a rather different kind of ring, namely a principal ideal domain, or PID. These rings, like semisimple rings, have the property that every (finit...

We prove that if a right distributive ring R, which has at least one completely prime ideal contained in the Jacobson radical, satisfies either a.c.c or d.c.c. on principal right annihilators, then the prime radical of R is the right singular ideal of R and is completely prime and nilpotent. These results generalize a theorem by Posner for right chain rings.

We show that standard arguments for deformations based on dimension counts can also be applied over a (not necessarily Noetherian) valuation ring A of rank 1. Key intermediate results are a principal ideal theorem for schemes of finite type over A, and a theorem on subadditivity of intersection codimension for schemes smooth over A.

A general variational principle of classical fields with a Lagrangian containing field quantity and its derivatives of up to the N-order is presented. Noether’s theorem is derived. The generalized Hamilton-Jacobi’s equation for the Hamilton’s principal functional is obtained. These results are surprisingly in great harmony with each other. They will be applied to general relativity in the subse...

recently, zhang and song [q. zhang, y. song, fixed point theory forgeneralized $varphi$-weak contractions,appl. math. lett. 22(2009) 75-78] proved a common fixed point theorem for two mapssatisfying generalized $varphi$-weak contractions. in this paper, we prove a common fixed point theorem fora family of compatible maps. in fact, a new generalization of zhangand song's theorem is given.

in this paper we study some results on noetherian semigroups. we  show that if  $s_s$ is an  strongly  faithful $s$-act and $s$ is a duo weakly noetherian, then we have the following.

binayak et al in [1] proved a fixed point of generalized kannan type-mappings in generalized menger spaces. in this paper we extend gen- eralized kannan-type mappings in generalized fuzzy metric spaces. then we prove a fixed point theorem of this kind of mapping in generalized fuzzy metric spaces. finally we present an example of our main result.