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.

an octahedral cobalt(iii) complex, trans-[(me2bpb)co(bzlan)2]clo4 (1), with h2me2bpb = n,n’-(4,5-dimethyl-1,2-phenylene)dipicolinamide and bzlan = benzylamine, has been synthesized and characterized by elemental analyses, ir, uv-vis, and 1h nmr spectroscopy. the structure of this complex has been determined by x-ray crystallography. the me2bpb2– is a di-anionic tetradentate ligand furnishing a ...

Let $R= \oplus_{ \alpha \in G} R_{\alpha}$ be a commutative ring with unity graded by an arbitrary grading monoid $G$. For each positive integer, the notions of graded-n-coherent module and are introduced. In this paper many results generalized from $n$-coherent rings to graded-$n$-coherent rings. last section, we provide necessary sufficient conditions for trivial extension graded-valuation ri...

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...