We obtain a covering theorem for E∗-dense E-semigroups showing that such a semigroup has an E∗-dense, strongly E∗-unitary E-semigroup as a cover and describe the structure of the latter semigroups.

in this article, we give several generalizations of the concept of annihilating ideal graph over a commutative ring with identity to modules. weobserve that over a commutative ring $r$, $bbb{ag}_*(_rm)$ isconnected and diam$bbb{ag}_*(_rm)leq 3$. moreover, if $bbb{ag}_*(_rm)$ contains a cycle, then $mbox{gr}bbb{ag}_*(_rm)leq 4$. also for an $r$-module $m$ with$bbb{a}_*(m)neq s(m)setminus {0}$, $...

All rings are commutative with identity and all modules are unital. Let R be a ring, M an R-module and R(M), the idealization of M . Homogeneuous ideals of R(M) have the form I(+)N where I is an ideal of R, N a submodule of M such that IM ⊆ N . In particular, [N : M ] (+)N is a homogeneous ideal of R(M). The purpose of this paper is to investigate how properties of the ideal [N : M ](+)N are re...

An elementary proof is given for Hutchinson’s duality theorem, which states that if a lattice identity λ holds in all submodule lattices of modules over a ring R with unit element then so does the dual of λ.

The authors study the dynamics of droplets quark-gluon plasma created in heavy iron collisions. relativistic quantum fluid undergoes Bjorken flow expansion and is modeled as conformal $N=4$ supersymmetric Yang-Mills plasma. In particular, they investigate holographic entropy production far from equilibrium where horizon area no longer a proper measure for but dynamic apparent assumed to be righ...

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...