in this paper we characterize the radical of an arbitrary‎ ‎submodule $n$ of a finitely generated free module $f$ over a‎ ‎commutatitve ring $r$ with identity‎. ‎also we study submodules of‎ ‎$f$ which satisfy the radical formula‎. ‎finally we derive‎ ‎necessary and sufficient conditions for $r$ to be a‎ ‎pr$ddot{mbox{u}}$fer domain‎, ‎in terms of the radical of a‎ ‎cyclic submodule in $rbigopl...

A code is -quasi-cyclic ( -QC) if there is an integer such that cyclic shift of a codeword by -positions is also a codeword. For = 1, cyclic codes are obtained. A dyadic code is a code which is closed under all dyadic shifts. An -QC dyadic ( -QCD) code is one which is both -QC and dyadic. QCD codes with = 1 give codes that are cyclic and dyadic (CD). In this correspondence, we obtain a simple c...

Dependencies between educational test items can be represented as quasi-orders on the item set of a knowledge domain and used for an efficient adaptive assessment of knowledge. One approach to uncovering such dependencies is by exploratory algorithms of item tree analysis (ITA). There are several methods of ITA available. The basic tool to compare such algorithms concerning their quality are la...