Let R be a commutative ring and G(R) be a graph with vertices as proper andnon-trivial ideals of R. Two distinct vertices I and J are said to be adjacentif and only if I + J = R. In this paper we study a graph constructed froma subgraph G(R)Δ(R) of G(R) which consists of all ideals I of R such thatI Δ J(R), where J(R) denotes the Jacobson radical of R. In this paper westudy about the relation b...

The Saturn ring plane crossings in 1995{96 allowed observers using the Hubble Space Telescope and the W. M. Keck telescope to image the planet's di use rings from 0.3 m { 2.2 m at a scattering angle 175 . We calculate the G ring re ectance for size distributions of dust to km-sized bodies derived from a physical, evolutionary model. The model tracks the evolution of the G ring from its initial ...

let r be a g-graded ring and m be a g-graded r-module. in this article, we introduce the concept of graded weakly classical prime submodules and give some properties of such submodules.