Journal of Algebra and Related Topics
Volume:2 Issue: 1, Summer 2014

In this paper, we introduce the notion of primal strong co-ideals and give some results involving them. It is shown that subtractive strong co-ideals are intersection of all primal strong co-ideals that contain them. Also we prove that the representation of strong co-ideals as reduced intersections of primal strong co-ideals is unique.

Let R be a commutative Noetherian ring and I be an ideal of R. We say that I satisfies the persistence property if AssR(R/Ik)⊆AssR(R/Ik) for all positive integers k≥1, which AssR(R/I) denotes the set of associated prime ideals of I. In this paper, we introduce a class of square-free monomial ideals in the polynomial ring R=K[x1,…,xn] over field K which are associated to unrooted trees such that if G is a unrooted tree and It(G) is the ideal generated by the paths of G of length t, then Jt(G):=It(G)∨, where I∨ denotes the Alexander dual of I, satisfies the persistence property. We also present a class of graphs such that the path ideals generated by paths of length two satisfy the persistence property. We conclude this paper by giving a criterion for normally torsion-freeness of monomial ideals.

Let M be a module over a commutative ring R and let N be a proper submodule of M. The total graph of M over R with respect to N, denoted by T(ΓN(M)), have been introduced and studied in [2]. In this paper, A generalization of the total graph T(ΓN(M)), denoted by T(ΓN,I(M)) is presented, where I is an ideal of R. It is the graph with all elements of M as vertices, and for distinct m,n∈M, the vertices m and n are adjacent if and only if m∈M(N,I), where M(N,I)={m∈M:rm∈N for some r∈R−I}. The main purpose of this paper is to extend the definitions and properties given in [2] and [12] to a more general case.

In this paper we study some results on Noetherian semigroups. We show that if SS is an strongly faithful S-act and Sis a duo weakly Noetherian, then we have the following.

A graph is \textit{symmetric}, if its automorphism group is transitive on the set of its arcs. In this paper, we classify all the connected cubic symmetric graphs of order 36p and 36p2, for each prime p , of which the proof depends on the classification of finite simple groups.

The notions of quasi-prime submodules and developed Zariski topology was introduced by the present authors in \cite{ah10}. In this paper we use these notions to define a scheme. For an R-module M, let X:={Q∈q\Spec(M)∣(Q:RM)∈\Spec(R)}. It is proved that (X,OX) is a locally ringed space. We study the morphism of locally ringed spaces induced by R-homomorphism M→N, and also by ring homomorphism R→S. Among other results, we show that (X,OX) is a scheme by putting some suitable conditions on M.