let $r$ be a commutative ring and let $m$ be an $r$-module. in this article, we introduce the concept of the zariski socles of submodules of $m$ and investigate their properties. also we study modules with noetherian second spectrum and obtain some related results.

Let $R$ be a commutative ring with identity and $M$ be an$R$-module. Let $FSpec(M)$ denotes the collection of all prime fuzzysubmodules of $M$. In this regards some basic properties of Zariskitopology on $FSpec(M)$ are investigated. In particular, we provesome equivalent conditions for irreducible subsets of thistopological space and it is shown under certain conditions$FSpec(M)$ is a $T_0-$spa...

×ØÖÖغ In the present paper, we present a slightly strengthened version of a well-known theorem of Belyi on the existence of " Belyi maps ". Roughly speaking, this strengthened version asserts that there exist Belyi maps which are unramified at [cf. Theorem 2.5] — or even near [cf. Corollary 3.2] — a prescribed finite set of points. Write C for the complex number field; Q ⊆ C for the subfield o...

For odd m, I write down tensors in C m ⊗C m ⊗C m of border rank at least 2m − 1, showing the non-triviality of the Young-flattening equations of [6] that vanish on the matrix multiplication tensor. I also study the border rank of the tensors of [1] and [3]. I show the tensors T 2 k ∈ C k ⊗C 2 k ⊗C 2 k , of [1], despite having rank equal to 2 k+1 − 1, have border rank equal to 2 k. I show the eq...