the rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. let $r$ be a ring. let $mathbb{a}(r)$ denote the set of all annihilating ideals of $r$ and let $mathbb{a}(r)^{*} = mathbb{a}(r)backslash {(0)}$. the annihilating-ideal graph of $r$, denoted by $mathbb{ag}(r)$  is an undirected simple graph whose vertex set is $mathbb{a}(r)...

‎Here‎, ‎we investigate a conjecture posed by Amiri and Ariannejad claiming‎ ‎that if every maximal subfield of a division ring $D$ has trivial normalizer‎, ‎then $D$ is commutative‎. ‎Using Amitsur classification of‎ ‎finite subgroups of division rings‎, ‎it is essentially shown that if‎ ‎$D$ is finite dimensional over its center then it contains a maximal‎ ‎subfield with non-trivial normalize...

We study the parity of the class number of the pth cyclotomic field for p prime. By analytic methods we derive a parity criterion in terms of polynomials over the field of 2 elements. The conjecture that the class number is odd for p a prime of the form 2q +1, with q prime, is proved in special cases, and a heuristic argument is given in favor of the conjecture. An implementation of the criteri...

Let e be a positive integer, p be an odd prime, q = p, and Fq be the finite field of q elements. Let f, g ∈ Fq[X,Y ]. The graph G = Gq(f, g) is a bipartite graph with vertex partitions P = F3q and L = F 3 q, and edges defined as follows: a vertex (p) = (p1, p2, p3) ∈ P is adjacent to a vertex [l] = [l1, l2, l3] ∈ L if and only if p2 + l2 = f(p1, l1) and p3 + l3 = g(p1, l1). Motivated by some qu...