Let R be a commutative ring. An R-module M is called co-multiplication provided that foreach submodule N of M there exists an ideal I of R such that N = (0 : I). In this paper weshow that co-multiplication modules are a generalization of strongly duo modules. Uniserialmodules of finite length and hence valuation Artinian rings are some distinguished classes ofco-multiplication rings. In additio...

We show that there exist noncommutative Ore extensions in which every right ideal is two-sided. This answers a problem posed by Marks in [5]. We also provide an easy construction of one sided duo rings.

We present an efficient implementation of ηT pairing on Intel Core 2 Duo processor. The processing speed of our implementation achieves 92 μsec over F 3 and 553 μsec over F 3 on 2.6GHz processor.