We explore the equimultiplicity theory of F-invariants Hilbert–Kunz multiplicity, F-signature, Frobenius Betti numbers, and Euler characteristic in strongly F-regular rings. Techniques introduced this paper provide a unified approach to study localization these invariants detection singularities.

A unitary strongly prime ring is defined as a prime ring whose central closure is simple with identity element. The class of unitary strongly prime rings is a special class of rings and the corresponding radical is called the unitary strongly prime radical. In this paper we prove some results on unitary strongly prime rings. The results are applied to study the unitary strongly prime radical of...

let $r$ be a commutative ring with identity. an ideal $i$ of a ring $r$is called an annihilating ideal if there exists $rin rsetminus {0}$ such that $ir=(0)$ and an ideal $i$ of$r$ is called an essential ideal if $i$ has non-zero intersectionwith every other non-zero ideal of $r$. thesum-annihilating essential ideal graph of $r$, denoted by $mathcal{ae}_r$, isa graph whose vertex set is the set...

Let $R$ be a commutative ring with identity and $N(R)$ and
 $J\left(R\right)$ denote the nilradical Jacobson radical
 of $R$, respectively. A proper ideal $I$ is called an
 n-ideal if for every $a,b\in R$, whenever $ab\in I$\ $a\notin
 N(R)$, then $b\in I$. In this paper, we introduce study
 J-ideals as new generalization n-ideals in rings.
 $I$\ $R$\ J-ideal $ab\i...

A simple, mild and efficient method for the regioselective nitration of various aromatic compounds usingpoly(4-vinylpyridinium nitrate) (P4VPyN) in the presence of melamine trisulfonic acid (MTSA), is reported.This method provides several advantages such as good selectiveity between ortho and para positions ofaromatic compounds, clean procedure, simple work-up, short reaction times, solvent-fre...