For two algebras $A$ and $B$, a linear map $T:A longrightarrow B$ is called separating, if $xcdot y=0$ implies $Txcdot Ty=0$ for all $x,yin A$. The general form and the automatic continuity of separating maps between various Banach algebras have been studied extensively. In this paper, we first extend the notion of separating map for module case and then we give a description of a linear se...

‎Let $p(z)$ be a polynomial of degree $n$ and for a complex number $alpha$‎, ‎let $D_{alpha}p(z)=np(z)+(alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $alpha$‎. ‎Dewan et al proved‎ ‎that if $p(z)$ has all its zeros in $|z| leq k, (kleq‎ ‎1),$ with $s$-fold zeros at the origin then for every‎ ‎$alphainmathbb{C}$ with $|alpha|geq k$‎, ‎begin{align*}‎ ‎max_{|z|=...

For a given monic polynomial p(t) of degree n over a commutative ring k, the splitting algebra is the universal k-algebra in which p(t) has n roots, or, more precisely, over which p(t) factors, p(t) = (t − ξ1) · · · (t − ξn). The symmetric group Sr for 1 ≤ r ≤ n acts on the splitting algebra by permuting the first r roots ξ1, . . . , ξr . We give a natural, simple condition on the polynomial p(...

The study of symmetry is one the most important and beautiful themes uniting various areas contemporary arithmetic. Algebraic structures are useful in pure mathematics for learning a geometrical object’s symmetries. In this paper, we introduce new concepts an algebraic structure called BCI-algebra, where present bipolar fuzzy (closed) BCI-positive implicative ideals BCI-commutative BCI-algebras...

We prove that any cycle Cn, n ≥ 4, with list assignment L, has a k -fold list coloring from the given lists if (i) each list contains at least 2k colors and (ii) Cn and L satisfy Hall’s condition for k -fold list colorings. Further, 2k in (i) cannot be replaced by 2k − 1 if either n is odd, or n is even and n ≥ k + 2. In other words, if n ≥ 4, the k -fold Hall number of a cycle Cn satifies h (C...

The main results: A ring R is CN if and only if for any x ∈ N(R) and y ∈ R, ((1+x)y)n+k = (1+x)n+kyn+k, where n is a fixed positive integer and k = 0, 1, 2; (2) Let R be a CN ring and n ≥ 1. If for any x, y ∈ R\N(R), (xy)n+k = xn+kyn+k, where k = 0, 1, 2, then R is commutative; (3) Let R be a ring and n ≥ 1. If for any x ∈ R\N(R) and y ∈ R, (xy)k = xkyk, k = n, n + 1, n + 2, then R is commutati...

a polynomial   1 2 ( , , , ) n f x x x is called multilinear if it is homogeneous and linear in every one of its variables. in the present paper our objective is to prove the following result: let   r be a prime k-algebra over a commutative ring   k with unity and let 1 2 ( , , , ) n f x x x be a multilinear polynomial over k. suppose that   d is a nonzero derivation on r such that ...

let $r$ be a commutative ring and let $m$ be an $r$-module. we define the small intersection graph $g(m)$ of $m$ with all non-small proper submodules of $m$ as vertices and two distinct vertices $n, k$ are adjacent if and only if $ncap k$ is a non-small submodule of $m$. in this article, we investigate the interplay between the graph-theoretic properties of $g(m)$ and algebraic properties of $m...

The notion of extended filter of a filter associated to a subset of Rl-monoids is defined and related properties are investigated. Rachunek and Salounova proved that positive implicative filters and Boolean filters in Rl-monoids coincide with a condition in Rachunek and Salounova (Acta Univ Palacki Olomuc Fac rer nat Math 48(1):93–107, 2009). In this paper, we prove that positive implicative fi...