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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 ...

The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in common. Lastly, the semisimple case (where the characteristic of the field is greater than 2) is considered and the minimum noncommutative counterexample to the Isomorphism Problem is identified.

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.

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.

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.

In this paper we focus on two generalizations of the notion of cyclicity of codes: polycyclic codes and sequential codes. We establish a duality between these two generalizations and also show connections between them and other well-known generalizations of cyclicity such as the notions of negacyclicity and constacyclicity. In particular, it is shown that a code $C$ is sequential and polycyclic if and only if $C$ and its dual C⊥ are both sequential if and only if $C$ and its dual C⊥ are both polycyclic. Furthermore, any one of these equivalent statements characterizes the family of constacyclic codes.

In this notice we describe the ideal structure of all cases of cyclic and negacyclic codes of length p s over a Galois ring alphabet that have not yet been discussed in the literature. Unlike in the cases reported earlier in the literature by various authors, the ambient spaces here are never chain rings. These ambient rings do nonetheless share the properties of being local and having a simple socle.

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.