somayeh hadjirezaei

Let $R$ be a commutative ring with identity. In this paper, we study 2-prime ideals of a Dedekind domain and a Pr\"{u}fer domain. We prove that a nonzero ideal $I$ of a Dedekind domain $R$ is 2-prime if and only if $I=P^{\alpha}$, for some maximal ideal $P$ of $R$ and positive integer $\alpha$. We give some results of ring $R$ in which every ideal $I$ is 2-prime. Finally, we define almost 2-prime, almost 2-primary and weakly 2-primary ideals, and investigate some properties of these ideals.

An R-module M is called Almost uniserial module, if any two non-isomorphic submodules of M are linearly ordered by inclusion. In this paper, we investigate some properties of Almost uniserial modules. We show that every finitely generated Almost uniserial module over a Noetherian ring, is torsion or torsionfree. Also the construction of a torsion Almost uniserial modules whose first nonzero Fitting ideal is a product of maximal ideals, is invetigated and torsion Almost uniserial modules over an integral domain and a UFD are characterized.

A finitely generated R-module is said to be a module of type (F r) if its (r−1)-th Fitting ideal is the zero ideal and its r-th Fitting ideal is a regular ideal. Let R be a commutative ring and N be a submodule of R n which is generated by columns of a matrix A=(a ij ) with a ij ∈R for all 1≤i≤n, j∈Λ, where Λ is a (possibly infinite) index set. Let M=R n /N be a module of type (F n−1) and T(M) be the submodule of M consisting of all elements of M that are annihilated by a regular element of R. For λ∈Λ, put M λ =R n /. The main result of this paper asserts that if M λ is a regular R-module, for some λ∈Λ, then M/T(M)≅M λ /T(M λ ). Also it is shown that if M λ is a regular torsionfree R -module, for some λ∈Λ, then M≅M λ . As a consequence we characterize all non-torsionfree modules over a regular ring, whose first nonzero Fitting ideals are maximal.

Let R be a commutative ring with identity and M be a finitely generated unital R-module. In this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. Moreover, we investigate some conditions which imply that the module M is the direct sum of some cyclic modules and free modules. Then some properties of Fitting ideals of modules which are the direct sum of finitely generated module and finitely generated multiplication module are shown. Finally, we study some properties of modules that are the direct sum of multiplication modules in terms of Fitting ideals.

In this paper we characterize all matrices of rank one over a unique factorization domain (UFD). Also we nd the R- module generated by the rows and the R-module generated by the columns of a matrix of rank one and assert some properties of them.

In this paper we characterize all 2×2 idempotent and nilpotent matrices over an integral domain and then we characterize all 2×2 strongly nil-clean matrices over a PID. Also, we determine when a 2×2 matrix over a UFD is nil-clean.