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We study the natural class of rings where each nilpotent element generates a nilpotent ideal, calling them the strongly[Formula: see text]-primal rings. We derive many basic properties of these rings, analyze their behavior under standard ring constructions and extensions, and taxonomize their relationship to other natural generalizations of commutativity. A slightly stronger condition is to assume that any nilpotent element generates a nilpotent ideal of the same index of nilpotence. We find that the difference between these two properties explains divergent behaviors in direct products, polynomial rings, Morita contexts, and other constructions.

The family of rings of the form \frac{\mathbb{Z}_{4}\left \langle x,y \right \rangle}{\left \langle x^2-a,y^2-b,yx-xy-2(c+dx+ey+fxy) \right \rangle} is investigated which contains the generalized Hamilton quaternions over $\Z_4$. These rings are local rings of order 256. This family has 256 rings contained in 88 distinct isomorphism classes. Of the 88 non-isomorphic rings, 10 are minimal reversible nonsymmetric rings and 21 are minimal abelian reflexive nonsemicommutative rings. Few such examples have been identified in the literature thus far. The computational methods used to identify the isomorphism classes are also highlighted. Finally, some generalized Hamilton quaternion rings over $\Z_{p^s}$ are characterized.

The classes of semicommutative, PS I, 2-primal, weakly-2-primal, NI, NR and Dedekind-finite rings form a strictly increasing chain of classes of rings. Examples have been provided by several authors to show most of the inclusions are strict. In this paper it is shown that every inclusion is strict. Furthermore, in most instances examples with the additional properties of being reflexive and/or abelian are constructed. In particular, an abelian nonreflexive NI non-weakly-2-primal ring is given, which shows that the classes of NI and weakly-2-primal rings differ; a fact that has been previously claimed without proof.

Characterizations of Morita context rings satisfying various additional ring properties, including reflexive, semiprime, prime, NI, weakly 2-primal and 2-primal, are given. The connections between ...

Background Chronic pain conditions are common among both male and female Iraq/Afghanistan-era veterans and can have substantial negative impacts on quality of life and function. Although in general women tend to report higher levels of pain intensity than men, findings remain mixed on whether gender differences in pain exist in Iraq/Afghanistan-era veterans. Additionally, the relationships between functional impairment, pain intensity, and gender remain unknown. Methods This project examined gender differences in pain intensity and pain interference in 875 male and female Iraq/Afghanistan-era veterans. Nonparametric Wilcoxon rank-tests examined gender differences in pain scores. Multivariable generalized linear regression modeling was used to evaluate the magnitude of pain intensity and interference across levels of chronicity and gender, and to evaluate the role of chronicity in gender effects in measures of pain and function. Results Pain intensity and interference scores were significantly greater among both male and female veterans reporting chronic pain relative to acute pain. Women veterans endorsed higher levels of pain intensity and pain interference compared with men. Results derived from multivariable analyses implicated pain intensity as a factor underlying gender differences in functional impairment among chronic pain sufferers, indicating that gender differences in functional measures were eliminated after controlling statistically for pain intensity. Conclusions Results demonstrate that the effects of functional impairment are impacted by pain intensity, and not by gender.

In a paper on the taxonomy of 2-primal rings, examples of various types of rings that are related to commutativity such as reduced, symmetric, duo, reversible and PS~I were given in order to show that the ring class inclusions were strict. Many of the rings given in the examples were infinite. In this paper, where possible, examples of minimal finite rings of the various types are given. Along with the rings in the previous taxonomy, NI, abelian and reflexive rings are also included.

It has recently been shown that a minimal reversible nonsymmetric ring has order 256 answering a question originally posed in a paper on a taxonomy of 2-primal rings. In a different work, questions on minimal rings having to do with this taxonomy were also answered. In this work it is shown that a minimal abelian reflexive nonsemicommutative ring also has order 256, a related question left open in these previous works. It is also shown here that [Formula: see text] is such a minimal ring. This is a consequence of the main result of the paper which is that a finite abelian reflexive ring of order [Formula: see text] for some prime [Formula: see text] and [Formula: see text] is reversible.

Codes over commutative Frobenius rings are studied with a focus on local Frobenius rings of order 16 for illustration. The main purpose of this work is to present a method for constructing a generating character for any commutative Frobenius ring. Given such a character, the MacWilliams identities for the complete and symmetrized weight enumerators can be easily found. As examples, generating characters for all commutative local Frobenius rings of order 16 are given. In addition, a canonical generator matrix for codes over local non-chain rings is discussed. The purpose is to show that when working over local non-chain rings, a canonical generator matrix exists but is less than useful which emphases the difficulties in working over such rings.

Let [math] be a sequence of [math] pairwise coprime positive integers, [math] , and [math] be a sequence of [math] different colors. Let [math] be an [math] matrix of colors in which row [math] consists of blocks of [math] consecutive entries of the same color with colors 0 through [math] repeated cyclically. The monochromatic column problem is to determine the number of columns of [math] in which every entry is the same color. The solution for a prime number of colors is provided.

Algebras having bases that consist entirely of units (called invertible algebras) are studied. Among other results, it is shown that all finite-dimensional algebras over a field other than the binary field F2 have this property. Invertible finite-dimensional algebras over F2 are fully characterized. Examples of invertible algebras are shown to include all (non-trivial) matrix rings over arbitrary algebras. In addition, various families of algebras, including group rings and crossed products, are characterized in terms of invertibility. Invertibility of infinite-dimensional algebras is also explored.

In this paper we study polycyclic codes of length $p^{s_1} \times \cdots \times p^{s_n}$\ over $\F_{p^a}$\ generated by a single monomial. These codes form a special class of abelian codes. We show that these codes arise from the product of certain single variable codes and we determine their minimum Hamming distance. Finally we extend the results of Massey et. al. in \cite{MASSEY_1973} on the weight retaining property of monomials in one variable to the weight retaining property of monomials in several variables.

Dual codes are defined with respect to non-degenerate sesquilinear or bilinear forms over a finite Frobenius ring. These dual codes have the properties one expects from a dual code: they satisfy a double-dual property, they have cardinality complementary to that of the primal code, and they satisfy the MacWilliams identities for the Hamming weight.

Marks showed that F 2 Q 8 , the F 2 group algebra over the quaternion group, is a reversible nonsymmetric ring, then questioned whether or not this ring is minimal with respect to cardinality. In this work, it is shown that the cardinality of a minimal reversible nonsymmetric ring is indeed 256. Furthermore, it is shown that although F 2 Q 8 is a duo ring, there are also examples of minimal reversible nonsymmetric rings which are nonduo.

article number 1750161; International audience; Given a finite ring $A$ which is a free left module over a subring $R$ of $A$, two types of $R$-bases are defined which in turn are used to define duality preserving maps from codes over $A$ to codes over $R$. The first type, pseudo-self-dual bases, are a generalization of trace orthogonal bases for fields. The second are called symmetric bases. Both types are illustrated with skew cyclic codes which are codes that are $A$-submodules of the skew polynomial ring $A[X;\theta]/\langle X^n-1\rangle$ (the classical cyclic codes are the case when $\theta=id$). When $A$ is commutative, there exists criteria for a skew cyclic code over $A$ to be self-dual. With this criteria and a duality preserving map, many self-dual codes over the subring $R$ can easily be found. In this fashion, numerous examples are given, some of which are not chain or serial rings.

We study codes over finite commutative local Frobenius rings of order 16. We define a standard form for the generator matrix for linear codes over these rings. Finally, we describe the generating characters for each ring which produce MacWilliams relations for codes over these rings.