the theorem 12 in [a note on $p$-nilpotence and solvability of finite groups, j. algebra 321(2009) 1555--1560.] investigated the non-abelian simple groups in which some maximal subgroups have primes indice. in this note we show that this result can be applied to prove that the finite groups in which every non-nilpotent maximal subgroup has prime index aresolvable.

Let $G$ be a monoid with identity $e$. In this paper, first we introduce the notions of $G$-graded hyperrings, graded hyperideals and graded hyperfields in the sense of Krasner hyperring $R$. Also, we define the notion of a greded $R$-hypermodules and some examples are presented. Then we investigate graded maximal, graded prime and graded primary hyperideals of a graded hyperring $R$. Finally, ...

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...

Recently Yehuda Rav has given the concept of Semi prime ideals in a general lattice by generalizing the notion of 0-distributive lattices. In this paper we study several properties of these ideals in a general nearlattice and include some of their characterizations. We give some results regarding maximal filters and include a number of Separation properties in a general nearlattice with respect...

In this paper, we introduce the notions of prime state filters, obstinate state filters, and primary state filters in state residuated lattices and study some properties of them. Several characterizations of these state filters are given and the prime state filter theorem is proved. In addition, we investigate the relations between them.

in this paper, we introduce a method by which we can find a close connection between the set of prime $z$-ideals of $c(x)$ and the same of $c(y)$, for some special subset $y$ of $x$. for instance, if $y=coz(f)$ for some $fin c(x)$, then there exists a one-to-one correspondence between the set of prime $z$-ideals of $c(y)$ and the set of prime $z$-ideals of $c(x)$ not containing $f$. moreover, c...

In this paper we enumerate fuzzy subgroups, up to a natural equivalence, of some finite abelian p-groups of rank two where p is any prime number. After obtaining the number of maximal chains of subgroups, we count fuzzy subgroups using inductive arguments. The number of such fuzzy subgroups forms a polynomial in p with pleasing combinatorial coefficients. By exploiting the order, we label the s...

in this paper we enumerate fuzzy subgroups, up to a natural equivalence, of some finite abelian p-groups of rank two where p is any prime number. after obtaining the number of maximal chains of subgroups, we count fuzzy subgroups using inductive arguments. the number of such fuzzy subgroups forms a polynomial in p with pleasing combinatorial coefficients. by exploiting the order, we label the s...

A ring is called a Gelfand ring (pm ring ) if each prime ideal is contained in a unique maximal ideal. For a Gelfand ring R with Jacobson radical zero, we show that the following are equivalent: (1) R is Artinian; (2) R is Noetherian; (3) R has a finite Goldie dimension; (4) Every maximal ideal is generated by an idempotent; (5) Max (R) is finite. We also give the following resu1ts:an ideal...