Sergio R. López-Permouth

Rings whose cyclics are essentially embeddable in projective modules

A ring R is quasi-Frobenius (QF) if and only if every right R-module is embeddable in a projective module. We call a ring R a right (left) CEPring if each cyclic right (left) R-module is essentially embeddable in a projective module. Examples of right CEP-rings include QF-rings and right uniserial rings. Indeed R is a QF-ring if and only if R is both a right and left CEP-ring [S]. A right CEP-ring which is QF-3 is shown to be QF (Theorem 3.3). Semiperfect CEP-rings and rings, each of whose homomorphic images is a right CEP-ring, are characterized in Theorems 5.2 and 6.2, respectively. The last section deals with split extensions of right uniserial rings as examples of CEP-rings.

On the equivalence of codes over rings and modules

In light of the result by Wood that codes over every finite Frobenius ring satisfy a version of the MacWilliams equivalence theorem, a proof for the converse is considered. A strategy is proposed that would reduce the question to problems dealing only with matrices over finite fields. Using this strategy, it is shown, among other things, that any left MacWilliams basic ring is Frobenius. The results and techniques in the paper also apply to related problems dealing with codes over modules.

On a class of QI-rings

0. Introduction. The concept of weak relative-injectivity of modules was introduced originally in [10], where it is shown that a semiperfect ring R is such that every cyclic right module is embeddable essentially in a projective right /?-module if and only if R is right artinian and every indecomposable projective right /?-module is uniform and weakly ft-injective. We show that in the above characterization the requirement that indecomposable projective right ft-modules be uniform is superfluous (Theorem 1.11). In this paper we further the study of weak relative-injectivity by considering the class of rings for which every right module is weakly injective relative to every finitely generated right module. We refer to such rings as right weakly-semisimple rings. The class of right weakly-semisimple rings includes properly all semisimple rings and is a subclass of the class of right Ql-rings. A ring R is said to be a right Ql-ring if every quasi-injective right /?-module is injective. Ql-rings have been studied in [2], [3], [4], [6], [7], [8], [11], among others. In [4], Boyle characterizes right Ql-rings as being those right noetherian rings for which every uniform cyclic right module is strongly prime. In contrast, we show that a ring R is right weakly-semisimple if and only if R is right noetherian and every finitely generated uniform right /^-module is compressible (Theorem 2.5). While it is not clear at this time if there exist any non-weakly-semisimple right Ql-rings, we show that a two-sided noetherian and hereditary ring is right weakly-semisimple if and only if it is right QI (Theorem 3.1). This implies that an example of a right Ql-ring which is not weakly-semisimple will necessarily be either a counter-example to Boyles conjecture or a right Ql-ring which is not left QI. Since a weakly /?-injective semiprime right Goldie ring must be left Goldie, we are able to apply some results of Kosler [11] to get that a right weakly-semisimple ring with restricted right minimum condition is left weakly-semisimple (Theorem 3.4). Throughout this paper all rings have 1 and all modules are unital right unless otherwise stated. The injective hull of the right /?-module M is denoted by E(MR) or simply E{M) if there is no ambiguity.

On categories of fuzzy modules

We show that for any ring R the category R-fzmod of fuzzy left R-modules is a top category and an additive category. R-fzmod has products, coproducts, kernels and cokernels, but it is not an abelian category. Projective, injective and free fuzzy left R-modules are characterized.

Continuous rings with ACC on essentials are Artinian

It is proved that a left continuous ring with ascending chain condition on essential left ideals is left artinian.

On the Equivalence of Codes over Finite Rings

Abstract.It is known that if a finite ring R is Frobenius then equivalences of linear codes over R are always monomial transformations. Among other results, in this paper we show that the converse of this result holds for finite local and homogeneous semilocal rings. Namely, it is shown that for every finite ring R which is a direct sum of local and homogeneous semilocal subrings, if every Hamming-weight preserving R-linear transformation of a codeC1 onto a code C2 is a monomial transformation then R is a Frobenius ring.

Characterizing rings in terms of the extent of the injectivity and projectivity of their modules

Abstract Given a ring R , we define its right i -profile (resp. right p -profile) to be the collection of injectivity domains (resp. projectivity domains) of its right R -modules. We study the lattice theoretic properties of these profiles and consider ways in which properties of the profiles may determine the structure of rings and vice versa. We show that the i -profile is isomorphic to an interval of the lattice of linear filters of right ideals of R , and is therefore modular and coatomic. In particular, we give a practical characterization of the profile of a right artinian ring and offer an example of a ring without injective left middle class for with the same is not true on the right-hand side. We characterize the p -profile of a right perfect ring and show through an example that the right p -profile of a ring is not necessarily a set. In addition, we use our results to provide a characterization of a special class of QF-rings in which the injectivity and projectivity domains of all modules coincide. The study of rings in terms of their ( i - or p -)profile was inspired by the study of rings with no right ( i - or p -)middle class, initiated in recent papers by Er, Lopez-Permouth and Sokmez, and by Holston, Lopez-Permouth and Orhan-Ertas.

Rings characterized by direct sums of cs modules

It is shown that a ring for which every CS right module is ∑CS is right artinian. As a consequence, it is also shown that over a ring R every direct sum of CS right R-modules is CS iff R is right artinian and the composition length of every uniform right R-module is at most 2.

Rings whose cyclics have finite Goldie dimension

Abstract A module M is said to be weakly injective if every finitely generated submodule of its injective hull E ( M ) is contained in a submodule X of E ( M ) isomorphic to M . We prove that a ring R satisfies the property that every cyclic right R -module has finite Goldie dimension if and only if every direct sum of (weakly) injective right R -modules is weakly injective. This is analog to the well-known characterization of right Noetherian rings as those for which direct sums of injective right modules are injective.

Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes

Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p^a,m) and generating sets for its ideals are considered. Along with some structure details of the ambient ring, the existance of a certain type of generating set for an ideal is proven.