in this paper, we introduce and study the prime, strongly prime,semiprime and irreducible fuzzy bi-ideals of a semigroup. we characterize thosesemigroups for which each fuzzy bi-ideal is semiprime. we also characterizethose semigroups for which each fuzzy bi-ideal is strongly prime.

let $g $ be a finite group and let $gamma(g)$ be the prime graph‎ ‎of g‎. ‎assume $2 < q = p^{alpha} < 100$‎. ‎we determine finite groups‎ ‎g such that $gamma(g) = gamma(u_3(q))$ and prove that if $q neq‎ ‎3‎, ‎5‎, ‎9‎, ‎17$‎, ‎then $u_3(q)$ is quasirecognizable by prime graph‎, ‎i.e‎. ‎if $g$ is a finite group with the same prime graph as the‎ ‎finite simple group $u_3(q)$‎, ‎then $g$ has a un...

Let R be a commutative ring with identity and M be a unitary R-module. Let : S(M) −! S(M) [ {;} be a function, where S(M) is the set of submodules ofM. Suppose n 2 is a positive integer. A proper submodule P of M is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 R and x 2 M and a1 . . . an−1x 2P(P), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x 2 P...

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. A proper submodule $P$ of $M$ is called strongly prime submodule if $(P + Rx : M)ysubseteq P$ for $x, yin M$, implies that $xin P$ or $yin P$. In this paper, we study more properties of strongly prime submodules. It is shown that a finitely generated $R$-module $M$ is Artinian if and only if $M$ is Noetherian and every st...

We investigate iterated schemata whose syntax integrates arithmetic parameters, indexed propositional variables (e.g. Pi), and iterated conjunctions/disjunctions (e.g. Vn i=1 Pi, where n is a parameter). Using this formalism gives us some extra information about the structure of the problem, which can be used in order to prove such a schema for all values of the parameters. The structure of the...

‎Let $G$ be a group with identity $e.$ Let $R$ be a $G$-graded‎ ‎commutative ring and $M$ a graded $R$-module‎. ‎In this paper‎, ‎we‎ ‎introduce several results concerning graded classical prime‎ ‎submodules‎. ‎For example‎, ‎we give a characterization of graded‎ ‎classical prime submodules‎. ‎Also‎, ‎the relations between graded‎ ‎classical prime and graded prime submodules of $M$ are studied‎.‎

River Ganga is one of the prime sacred National Rivers of India, closely associated with economic, social, and cultural heritage of Indian people. Recently, it has been subjected to immense degradation and pollution as a result of receiving huge amounts of domestic and industrial wastewater as well as religious ritual activities and surface runoff. The present study attempts to study spatial an...

We introduce the notion of ideal, prime ideal, lter, fuzzy ideal, fuzzy prime ideal, fuzzy lter of ordered $Gamma$-semiring and study their properties and relations between them. We characterize the prime ideals and lters of ordered $Gamma$-semiring with respect to fuzzy ideals and fuzzy l- ters respectively.

River Ganga is one of the prime sacred National Rivers of India, closely associated with economic, social, and cultural heritage of Indian people. Recently, it has been subjected to immense degradation and pollution as a result of receiving huge amounts of domestic and industrial wastewater as well as religious ritual activities and surface runoff. The present study attempts to study spatial an...