A ring R is called right zip provided that if the annihilator r_R(X) of a subset X of R is zero,then r_R(Y)=0 for some finite subset Y■X.Such rings have been studied in literature.For a right R-module M,we introduce the notion of a zip module,which is a generalization of the right zip ring.A number of properties of this sort of modules are established,and the equivalent conditions of the right zip ring R are given.Moreover,the zip properties of matrices and polynomials over a module M are studied.
For a right R-module N, we introduce the quasi-Armendariz modules which are a common generalization of the Armendariz modules and the quasi-Armendariz rings, and investigate their properties. Moreover, we prove that NR is quasi-Armendariz if and only if Mm(N)Mm(R) is quasi-Armendariz if and only if Tm(N)Tm(R) is quasi-Armendariz, where Mm(N) and Tm(N) denote the m×m full matrix and the m×m upper triangular matrix over N, respectively. NR is quasi-Armendariz if and only if N[x]R[x] is quasi-Armendariz. It is shown that every quasi-Baer module is quasi-Armendariz module.
In this note we first show that if H is a finite-dimensional Hopf algebra in a group Yetter-Drinfel'd category LLYD(π) over a crossed Hopf group-coalgebra L, then its dual H is also a Hopf algebra in the category LLYD(π). Then we establish the fundamental theorem of Hopf modules for H in the category LLYD(π).
