let r be a commutative ring with identity and m be a unitary r-module. let : s(m) −! s(m) [ {;} be a function, where s(m) is the set of submodules ofm. suppose n  2 is a positive integer. a proper submodule p of m is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 r and x 2 m and a1 . . . an−1x 2p(p), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x...

The influence of benzoate anions on the corrosion inhibition effect of benzotriazole (BTA) for mild steel in sulfuric acid has been investigated. The investigation was carried out by electrochemical impedance spectroscopy (EIS) and potentiodynamic polarization techniques. The addition of benzoate ions greatly enhances the inhibition efficiency of BTA in a synergistic manner. This effect has bee...

A ring $R$ with an automorphism $sigma$ and a $sigma$-derivation $delta$ is called $delta$-quasi-Baer (resp., $sigma$-invariant quasi-Baer) if the right annihilator of every $delta$-ideal (resp., $sigma$-invariant ideal) of $R$ is generated by an idempotent, as a right ideal. In this paper, we study Baer and quasi-Baer properties of skew PBW extensions. More exactly, let $A=sigma(R)leftlangle x...

In the combination of 5-fluorophenyl-1, 3, 7, 9-tetra methyl pyrido [2, 3, d, d, 5, 6 di pyrimidine -2, 4, 6, and 8 tetron, there are 4 rings, respectively, ring that is attached to the halogen ring A and, respectively, a ring The lower and the right to the left of the ring A are called C, B, and D. Calculations and studies show that the amount of aromatics in halogen-ring A in all cases has th...

We give a criterion for when idempotents of a ring R which commute modulo the Jacobson radical J(R) can be lifted to commuting idempotents of R. If such lifting is possible, we give extra information about the lifts. A “half-commuting” analogue is also proven, and this is used to give sufficient conditions for a ring to have the internal exchange property. In particular, we show that if R/J(R) ...

introduction:  stereotactic  radiosurgery  is  a  technique  for  the  treatment  of  intracranial  lesions  with  highly  collimated  x-ray  beam.  linac-based  radiosurgery  is  currently  performed  by  two  techniques:  couch or pedestal mounted. in the first technique, a device is required to affix patient's head to a couch  and  moreover  to  translate  it  accurately.  structure  of...