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...

We report a set of rare copper-catalyzed reactions of alkanes with simple amides, sulfonamides, and imides (i.e., benzamides, tosylamides, carbamates, and phthalimide) to form the corresponding N-alkyl products. The reactions lead to functionalization at secondary C-H bonds over tertiary C-H bonds and even occur at primary C-H bonds. [(phen)Cu(phth)] (1-phth) and [(phen)Cu(phth)2] (1-phth2), wh...

Antioxidants are vital substances which possess the ability to protect the body from damage caused by free radical induced oxidative stress. A variety of free radical scavenging antioxidants exist within the body which many of them are derived from dietary sources like fruits, vegetables and teas. In this study the antioxidant activity and radical scavenging activity of methanolic extracts of s...

We present a case of cervical necrotising fasciitis in a 56 year old man, secondary to a rare mastoid infection. The patient had coexisting diabetes mellitus and hypertension. He was treated with early surgical debridement followed by neck and chest reconstruction and radical mastoidectomy. Aggressive antibiotic therapy and supportive care was given. He recovered well with minimal residual func...

Under mechanical stress, the hexaarylbiimidazole (HABI) motif can cleave to triphenylimidazolyl radicals when incorporated into a polymer matrix. The mechanically produced coloured radicals can initiate secondary radical reactions yielding polymer networks. Thus, the HABI mechanophore combines optical reporting of bond scission and reinforcement of polymers in a single molecular moiety.

A synthetic approach to highly functionalized cyclopentanes that incorporate adjacent secondary and tertiary alcohols is described. These systems, prepared in one step by a ketyl radical cyclization of β-disubstituted acrylates, represent the core of the marine prostanoid family of natural products.

Let S-Set be the category of $S$-sets, sets together with the actions of a semigroup $S$ on them. And, let S-Pos be the category of $S$-posets, posets together with the actions compatible with the orders on them. In this paper we show that the category S-Pos is a radical extension of S-Set; that is there is a radical on the category S-Pos, the order desolator radical, whose torsion-free class i...