In this paper, we introduce the notion of the (α, β)-weakly smooth fuzzy continuous proper function and discuss its properties. We also study several notions of connectedness in smooth fuzzy topological spaces and establish that the product of connected sets (spaces) is not connected in any sense, as well as investigate continuous images of smooth connected sets (spaces) under (α, β)-weakly smooth fuzzy continuous functions.
We establish some properties of the class of order weakly compact operators on Banach lattices. As consequences, we obtain some characterizations of Banach lattices with order continuous norms or whose topological duals have order continuous norms.
Let R be a commutative ring. The annihilator graph of R, denoted by AG(R), is the undirected graph with all nonzero zero-divisors of R as vertex set, and two distinct vertices x and y are adjacent if and only if ann(xy) \neqann R(x)\cup annR(y), where for z \in R, annR(z) = {r \in R: rz = 0}. In this paper, we characterize all finite commutative rings R with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings R whose annihilator graphs have clique number 1, 2 or 3. Also, we investigate some properties of the annihilator graph under the extension of R to polynomial rings and rings of fractions. For instance, we show that the graphs AG(R) and AG(T(R)) are isomorphic, where T(R) is the total quotient ring of R. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo n, where n>1., Mojgan Afkhami, Kazem Khashyarmanesh, Zohreh Rajabi., and Obsahuje bibliografii
Let $R$ be a commutative Noetherian ring, $\mathfrak {a}$ an ideal of $R$, $M$ an $R$-module and $t$ a non-negative integer. In this paper we show that the class of minimax modules includes the class of $\mathcal {AF}$ modules. The main result is that if the $R$-module ${\rm Ext}^t_R(R/\mathfrak {a},M)$ is finite (finitely generated), $H^i_\mathfrak {a}(M)$ is $\mathfrak {a} $-cofinite for all $i<t$ and $H^t_\mathfrak {a}(M)$ is minimax then $H^t_\mathfrak {a}(M)$ is $\mathfrak {a} $-cofinite. As a consequence we show that if $M$ and $N$ are finite $R$-modules and $H^i_\mathfrak {a}(N)$ is minimax for all $i<t$ then the set of associated prime ideals of the generalized local cohomology module $H^t_\mathfrak {a}(M,N)$ is finite.
Let $R$ be a commutative Noetherian ring with identity and $I$ an ideal of $R$. It is shown that, if $M$ is a non-zero minimax $R$-module such that $\dim \mathop {\rm Supp} H^i_I (M) \leq 1$ for all $i$, then the $R$-module $H^i_I(M)$ is $I$-cominimax for all $i$. In fact, $H^i_I(M)$ is $I$-cofinite for all $i\geq 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $\dim \mathop {\rm Supp} H^i_I (M)\leq 2$ for all $i<t$, then ${\rm Ext}^j_R (R/I, H^i_I (M))$ and ${\rm Hom}_R(R/I, H^t_I(M))$ are weakly Laskerian for all $i<t$ and all $j \geq 0$. As a consequence, the set of associated primes of $H^i_I (M)$ is finite for all $i\geq 0$, whenever $\dim R/I \leq 2$ and $M$ is weakly Laskerian.
In this paper we obtain all solutions which depend only on $r$ for a class of partial differential equations of higher order with singular coefficients.
We prove identities involving sums of Legendre and Jacobi polynomials. The identities are related to Green’s functions for powers of the invariant Laplacian and to the Minakshisundaram-Pleijel zeta function.