In this paper, we introduce the class of ideals with (d1; : : :; dm)-linear quotients generalizing the class of ideals with linear quotients. Under suitable conditions, we control the numerical invariants of a minimal free resolution of ideals with (d1; : : :; dm)-linear quotients. In particular, we show that their  rst module of syzygies is a componentwise linear module.

Let  be a set of primes, and let G be a  nite -separable group. We consider the Isaacs B-characters. We show that if N is a normal subgroup of G, then B (G=N) = Irr(G=N) \ B (G).

Using a recently obtained criterion of C*-reflexivity for commutative C*-algebras, we show that the C*-algebra of continuous functions on the Higson corona is not C*-reflexive. This implies that C*-reflexivity doesn’t pass to quotient C*-algebras.

Let $R$ be a commutative ring with identity, $M$ be a unital $R$-module and let $L$ be a complete Heyting algebra. In this paper, among results on colon structures of $L$-neutrosophic submodules and $L$-neutrosophic ideals, we introduce and study the notion of primary (and prime) $L$-neutrosophic submodules and give connections with primary (prime) behavior of its $t$, $i$ and $f$ components. Then, for a multiplicatively closed subset $S$ of $R$, we define the notion of localization formation for an $L$-neutrosophic submodule $\lambda$ of $M$ and study its behavior. Some types of $L$-neutrosophic quotients will also be investigated.

Background: Food insecurity refers to the lack of sufficient and safe availability of the food necessary for a healthy and active life. Intelligence is the general talent of a person for understanding the world and meeting his/ her expectations. Many studies have suggested anassociation between a person’s nutrition and his/her mental abilities and intelligence quotient.Methods: The current research was a case-control in which 222 female students aged 9-11 years (111 with low intelligence quotients as the case group and 111 with average intelligence quotients as the control group) were randomly chosen from public girls primary schools in Bandar Anzali, Iran. General and demographic characteristics were gathered, and USDA household food insecurity questionnaires were completed by the students mothers. Wechslers revised intelligence test was used to determine each student’s intelligence quotient. The acquired data was analyzed in chi-square 2, student ttest, SPSS (16.0 version), and Stata 11 SE.Findings: The rate of food insecurity in this study was 51%. Food insecurity was found to have a significant association to the students intelligence quotients (p>0.0001). Food insecurity in case and control group households were 58.6% and 22.5%, respectively, and the average intelligence quotient was 77.97±5.56 for students in the case group and 94.6±5.47 for children in the control group. Furthermore, the association between food insecurity and mothers employment, parents educational levels, household economic level, and number of employed household members were also found to be statistically significant.conclusion: The association between food insecurity and students intelligence quotients was completely significant (p<0.0001), and the rate of food insecurity in case group households was 2.6 times more than of control group households. Therefore, it is necessary to pay more attention to household food security and its consequences, including childrens low intelligence quotients.

Groups associated to surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called ramification structure. Gül and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, admit ramification structures. We extend their result by showing that quotients of generalisations of the GGS-groups, namely multi-EGS groups, also admit ramification structures.

The eff, ect of restricting the set of primes dividing the orders of the , nite quotients of a group is investigated. Particular attention is paid to abelian, soluble, locally soluble and locally , nite groups. The connection with the extraction of roots is explored.

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and$I\subset S$ be a monomial ideal with a linearresolution. Let$\frak{m}=(x_1,\ldots,x_n)$ be the unique homogeneous maximal ideal and $I\frak{m}$ be apolymatroidal ideal. We prove that if either $I\frak{m}$ is polymatroidal with strongexchange property, or $I$ is a monomial ideal in at most 4variables, then $I$ is polymatroidal. We also show that the firsthomological shift ideal of polymatroidal ideal is againpolymatroidal.

‎In this article‎, ‎we first introduce the concept of $z$-sets in the ring $ \mathcal{R}(L)$ of real-valued continuous functions on a completely regular frame $L$‎, ‎and give some properties of them‎. ‎Let $S^{-1}_{\mathcal{F}}\mathcal{R}(L)$ denote‎ ‎the ring of fractions of the ring $ \mathcal{R}(L)$‎, ‎where ${\mathcal{F}}$ is a ${\mathrm{coz}}$-filter on $L$ and‎ ‎$S_{\mathcal{F}}$ is a multiplicatively closed subset related to ${\mathcal{F}}$‎. ‎We show that $S^{-1}_{\mathcal{F}}\mathcal{R}(L)$‎ ‎may be realized as the direct limits of the subrings $\mathcal{R}(A)$‎, ‎where‎ ‎$A\in‎ \{‎\mathfrak{o}_L\big({\mathrm{coz}}(\alpha)\big)‎ ‎\colon \alpha\in S_{\mathcal{F}}\}$‎. ‎Also‎, ‎we show that ${\mathrm{Q_{cl} }}\mathcal{R}(L)=\mathcal{R}(L)$, ‎if and only if‎ ‎$\mathcal{R}(L)$ is a special saturated ring‎.

In this paper at first we show that the cosets of a fuzzy ideal $\mu$ in a BCK–algebra $X$ form another BCK–algebra $frac{\mu}{X}$ called the fuzzy quotient BCK–algebra of X by $\mu$. Also we show that $frac{\mu}{X}$ is a fuzzy partition of $X$, and some isomorphism theorems are proved. Moreover we prove that, if the associated fuzzy similarity relation of a fuzzy partition $P$ of a commutative BCK–algebra is compatible, then $P$ is a fuzzy quotient BCK–algebra. Finally the notion of a coset of a fuzzy ideal and an element of a BCK-algebra is defined and some related theorems are proved.