Narrow Results

Let $R$ be a ∗-ring. Then $R$ is called a ∗-symmetric ring if for any $a,b,c\in R,abc=0$ implies $ac{b}^{\ast }=0.$ Obviously, ∗-symmetric ring is certainly symmetric ring, while ∗-ring and symmetric ring does not equal ∗-symmetric ring. Even though reduced rings are symmetric, reduced rings need not be ∗-symmetric ring. In this paper, we further add some conditions for symmetric ring to be ∗-symmetric and reduced. As an application, we discover if $R$ be a ∗-symmetric ring, then $R^{\#}=R^{EP}=R^{+}=R^{\mathrm{\text{reg}}}.$ Also, we expand the definition of ∗-symmetric ring to different combinations of $a,b,c$. The properties of ∗-symmetric ring play a role in generalized inverses of elements in rings.

A symmetric ring is reversible and a reversible ring is abelian reflexive. It has been shown that the smallest order for both a reversible nonsymmetric ring and an abelian reflexive nonreversible is 256. Examples of such rings are ${𝔽}_2{Q}_8$ and ${𝔽}_2{D}_8$, respectively. The aim of the paper is to show that for a given prime $p$, the minimum $k$ such that there exists a reversible nonsymmetric ring of order $p^k$ is 7 when $3|p-1$ and 8 when $3\nmid p-1$. It is also established that for a given prime $p$, the minimum $k$ such that there exists an abelian reflexive nonreversible ring of order $p^k$ is 8. Thus, an abelian reflexive ring of order $p^k$ is symmetric, when $k<7$, or when $k=7$ and $3\nmid p-1$.

The study of symmetric rings has important roles in ring theory and module theory. We investigate the structure of ring properties related to symmetric rings and introduce H-symmetric and π-symmetric as generalizations. We construct a non-symmetric reversible ring whose basic structure is infinite-dimensional, comparing with the finite-dimensional such rings of Anderson, Camillo and Marks. The structure of π-reversible rings (with or without identity) of minimal order is completely investigated. The properties of zero-dividing polynomials over IFP rings are studied more to show that polynomial rings over symmetric rings are π-symmetric. It is also proved that all conditions in relation with our arguments in this paper are equivalent for regular or locally finite rings.

This paper concerns several ring theoretic properties related to matrices and polynomials. The basic properties of $\pi$-reversible and power-Armendariz are studied. We provide a method by which one can always construct a power-Armendariz ring but neither symmetric nor Armendariz from given any symmetric ring. We investigate next various interesting relations among ring theoretic properties containing $\pi$-reversibility and power-Armendariz condition.

The theory of linear codes over finite fields has been extended by A. Nechaev to codes over quasi-Frobenius modules over commutative rings, and by J. Wood to codes over (not necessarily commutative) finite Frobenius rings. In the present paper, we subsume these results by studying linear codes over quasi-Frobenius and Frobenius modules over any finite ring. Using the character module of the ring as alphabet, we show that fundamental results like MacWilliams' theorems on weight enumerators and code isometry can be obtained in this general setting.

It was shown by Galovich that if $R$ is a commutative unique factorization ring (UFR) with identity, then $R$ is a local ring with a nil maximal ideal. In this paper, we generalize Galovich’s results to the non-commutative case.

The aim of this paper is to investigate the interplay between the algebraic properties of a skew Poincaré–Birkhoff–Witt extesion ring $A=\sigma (R)〈x_1,\dots ,x_n〉$ and the graph-theoretic properties of its zero-divisor graph. We are interested in studying the diameter of the zero-divisor graph of skew PBW extension rings. Among other results, we give a complete characterization of the possible diameters of $\mathrm{\Gamma }(A)$ in terms of the diameter of $\mathrm{\Gamma }(R)$.

The focus of this paper is on a ring construction H_n(R; σ) based on a given ring R and a Hochschild 2-cocycle σ. This construction is a unified generalization of the ring R[x]/(xⁿ⁺¹) and the Hochschild extension H_σ(R, R). Here we discuss when the ring H_n(R; σ) is reversible, symmetric, Armendariz, abelian and uniquely clean, respectively. Several known results of R[x]/(xⁿ⁺¹) and H_σ(R, R) are extended to H_n(R; σ), and new examples of reversible, symmetric and Armendariz rings are given.