F. J. Lobillo

Prime fuzzy ideals over noncommutative rings

In this paper we introduce prime fuzzy ideals over a noncommutative ring. This notion of primeness is equivalent to level cuts being crisp prime ideals. It also generalizes the one provided by Kumbhojkar and Bapat in 1993, which lacks this equivalence in a noncommutative setting. Semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes. This allows us to introduce the fuzzy prime radical and contribute to establish the basis of a Fuzzy Noncommutative Ring Theory.

A New Perspective of Cyclicity in Convolutional Codes

In this paper, we propose a new way of providing cyclic structures to convolutional codes. We define the skew cyclic convolutional codes as left ideals of a quotient ring of a suitable non-commutative polynomial ring. In contrast to the previous approaches to cyclicity for convolutional codes, we use Ore polynomials with coefficients in a field (the rational function field over a finite field), so their arithmetic is very well known and we may proceed similarly to cyclic block codes. In particular, we show how to obtain easily skew cyclic convolutional codes of a given dimension, and we compute an idempotent generator of the code and its dual.

Homological Computations in PBW Modules

In this paper the Poincaré–Birkhoff–Witt (PBW) rings are characterized. Gröbner bases techniques are also developed for these rings. An explicit presentation of Exti(M,N) is provided when N is a centralizing bimodule.

RE-FILTERING AND EXACTNESS OF THE GELFAND-KIRILLOV DIMENSION

Abstract We prove that any multi-filtered algebra with semi-commutative associated graded algebra can be endowed with a locally finite filtration keeping up the semi-commutativity of the associated graded algebra. As consequences, we obtain that Gelfand–Kirillov dimension is exact for finitely generated modules and that the algebra is finitely partitive. Our methods apply to algebras of current interest like the quantized enveloping algebras, iterated differential operators algebras, quantum matrices or quantum Weyl algebras.

Auslander-Regular and Cohen–Macaulay Quantum Groups

The quantized enveloping C(q)-algebra Uq(C) associated to a Cartan matirx C is Auslander-regular and Cohen–Macaulay. This is deduced from a general theorem, which also applies to solvable polynomial algebras. The results are obtained by constructing a new filtration keeping the properties of the associated graded algebra of a given multi-filtered algebra.

Ideal Codes Over Separable Ring Extensions

In this paper, an application of the theoretical algebraic notion of a separable ring extension in the realm of cyclic convolutional codes or, more generally, ideal codes, is investigated. It is worked under very mild conditions that cover all previously known as well as new non-trivial examples. It is proved that ideal codes are direct summands, as left ideals, of the underlying non-commutative algebra, in analogy with cyclic block codes. This implies, in particular, that they are generated by a non-commutative idempotent polynomial. Hence, by using a suitable separability element, an efficient algorithm for computing one of such idempotents is designed. We show that such an idempotent generator polynomial can be used to get information on the free distance of the convolutional code.

PRIMALITY TEST IN ITERATED ORE EXTENSIONS

We give a primality test for two-sided ideals over rings belonging to a class of iterated Ore extensions of a field, which includes differential operators rings and coordinate rings of quantum affine spaces. When applied to ideals of commutative polynomial rings, the test boils down to the given in (Gianni et al. J. Symb. Comput. 1988, 6, 149–167).

A Sugiyama-Like Decoding Algorithm for Convolutional Codes

We propose a decoding algorithm for a class of convolutional codes called skew Reed-Solomon convolutional codes. These are convolutional codes of designed Hamming distance endowed with a cyclic structure yielding a left ideal of a non-commutative ring (a quotient of a skew polynomial ring). In this setting, right and left division algorithms exist, so our algorithm follows the guidelines of the Sugiyama’s procedure for finding the error locator and error evaluator polynomials for Goppa block codes.

Separable Automorphisms on Matrix Algebras over Finite Field Extensions: Applications to Ideal Codes.

Let (F ⊆ K) an extension of finite fields and (A = Mn K) be the ring of square matrices of order n over (K) viewed as an algebra over (F). Given an (F)--automorphism (σ) on (A) the Ore extension (A[z;σ]) may be used to built certain convolutional codes, namely, the ideal codes. We provide an algorithm to decide if the automorphism (σ) on (A) is a separable returning the corresponding separability element (p). In this case (p) is also a separability element for the extension (F[z] ⊆ A[z;σ]), and as a consequence ideal codes are generated by idempotents in (A[z;σ]), which can be computed applying previous algorithms of the authors.

Computing the bound of an Ore polynomial. Applications to factorization

We develop a fast algorithm for computing the bound of an Ore polynomial over a skew field, under mild conditions. As an application, we state a criterion for deciding whether a bounded Ore polynomial is irreducible, and we discuss a factorization algorithm. The asymptotic time complexity in the degree of the given Ore polynomial is studied. In the class of Ore polynomials over a finite field, our algorithm is an alternative to Giesbretchs one that reduces the complexity in the degree of the polynomial.