The Zero Divisor Graphs of Commutative Local Rings of Order p 4 and p 5

To each commutative ring R we can associate a zero divisor graph whose vertices are the zero divisors of R and such that two vertices are adjacent if their product is zero. Detecting isomorphisms among zero divisor graphs can be reduced to the problem of computing the classes of R under a suitable semigroup congruence. Presently, we introduce a strategy for computing this quotient for local rings using knowledge about a generating set for the maximal ideal. As an example, we then compute Γ(R) for several classes of rings; with the results in [4 Bloomfield , N. , Wickham , C. ( 2010 ). Local rings with genus 2 zero divisor graph . Comm. Alg. 38 ( 8 ): 2965 – 2980 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] these classes include all local rings of order p ⁴ and p ⁵ for prime p.     

Ideal-Symmetric and Semiprime Rings

Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal A of a ring R symmetric if rst ∈ A implies rts ∈ A for r, s, t ∈ R. R is usually called symmetric if 0 is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process, we introduce the concept of an ideal-symmetric ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring R[x] over an ideal-symmetric ring R need not be ideal-symmetric, but it is shown that the factor ring R[x]/xⁿR[x] is ideal-symmetric over a semiprime ring R.     

Symmetric Powers of Nat 𝔰𝔩2(𝕂)

We identify the spaces of homogeneous polynomials in two variables 𝕂[Y^k, XY^k?1, ?, X^k] among representations of the Lie ring 𝔰𝔩₂(𝕂). This amounts to constructing a compatible 𝕂-linear structure on some abstract 𝔰𝔩₂(𝕂)-modules, where 𝔰𝔩₂(𝕂) is viewed as a Lie ring.     

The Exchange Property in Grothendieck Categories: Applications

We introduce the concept of an object with the (finite) exchange property in an arbitrary Grothendieck category, and we present the basic properties of such an object. Applications are given for categories of graded modules and for categories of comodules over a coalgebra. Among other results, it is proved that an arbitrary coalgebra 𝒞 over a field has the finite exchange property.     

Group Rings Whose Symmetric Units are Solvable

Let K be an infinite field of characteristic different from 2, and G a group. Under suitable restrictions upon G, we classify the groups such that the symmetric units of KG satisfy the solvability identity (x ₁, x ₂,…, x _{2 ⁿ} ) ^o  = 1, for some n.     

A Note on the Dimension Theory of ℤ n -Graded Rings

In this note we want to generalize some of the results in [1 Brewer , J. , Montgomery , P. , Rutter E. , Heinzer , W. ( 1973 ). Krull dimension of polynomial rings in “Conference on Commutative Algebra, Lawrence 1972.” . Springer Lecture Notes in Mathematics 311 : 26 – 45 .[Crossref] , [Google Scholar]] from polynomial rings in several indeterminates to arbitrary ? ⁿ -graded commutative rings. We will prove an analogue of Jaffard's Special Chain Theorem and a similar result for the height of a prime ideal 𝔭 over its graded core 𝔭*.     

Maps Preserving Zeros of xy*

For a prime ring A with involution, we explore the characterization of additive bijective maps φ: A → A such that φ(x)φ(y)* = 0 whenever xy* = 0. In particular, we show that if A is prime, unital, and generated by nontrivial idempotents, then there is a *-monomorphism g of A into Q, and an element r ∈ A such that φ(x) = rg(x) for all x ∈ A.     

Effect of Rashba spin-orbit coupling on electron transport in asymmetrically coupled regular polygonal quantum ring

The effect of Rashba spin-orbit coupling (SOC) on electron transport in asymmetrically coupled regular polygonal quantum ring is investigated. In absence of SOC, two kinds of conductance zeros appear periodically. In presence of SOC, one kind of conductance zero can be lifted by the Rashba SOC, the others persist.