We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generator sets of its quasi-injective hull. Several consequences are deduced. In particular, it is shown that every right hereditary module with countably generated quasi-injective hull is noetherian. Or that every right hereditary ring with finitely generated injective hull is artinian, thus answe...

The design of an efficient code-based signature scheme is by all means still an open problem. In this paper, we propose a simple and efficient scheme following a framework introduced by Lyubashevsky to construct a digital signature scheme from an identification scheme. The scheme is based on quasi-cyclic codes and, while security relies on the ring algebra that is associated with them, the prop...

We prove an analogue of Lowrey--Sch\"urg's algebraic Spivak's theorem when working over a base ring $A$ that is either field or nice enough discrete valuation ring, and after inverting the residual characteristic exponent $e$ in coefficients. By this result bordism groups quasi-projective derived $A$-schemes can be generated by classical cycles, leading to vanishing results for low degree $e$-i...

By means of a kind of new idea, we redefine some kinds of fuzzy ideals in a ring and investigate some of their related properties. The concepts of strong prime (semiprime) generalized fuzzy (bi-, quasi-) ideals in rings are introduced. In particular, we discuss the relationships between strong prime (resp., semiprime) generalized fuzzy (bi-, quasi-) ideals and prime (resp. semiprime) generalize...

In this paper, we study the structure of 1-generator quasi-cyclic codes over the ring R = F2 +uF2 + vF2 + uvF2, with u2 = v2 = 0 and uv = vu. We determine the minimal spanning sets for these codes. As a generalization of these codes, we also investigate the structure of 1-generator generalized quasi-cyclic codes over R and determine a BCH type bound for them.

Keigher showed that quasi-prime ideals in differential commutative rings are analogues of prime ideals in commutative rings. In that direction, he introduced and studied new types of differential rings using quasi-prime ideals of a differential ring. In the same sprit, we define and study two new types of differential rings which lead to the mirrors of the corresponding results on von Neumann r...

Generalized quasi-cyclic (GQC) codes with arbitrary lengths over the ring Fq + uFq, where u 2 = 0, q = p, n a positive integer and p a prime number, are investigated. By the Chinese Remainder Theorem, structural properties and the decomposition of GQC codes are given. For 1-generator GQC codes, minimal generating sets and lower bounds on the minimum distance are given. As a special class of GQC...

We give a new representation theoretic interpretation of the ring of quasisymmetric functions. This is obtained by showing that the super analogue of the Gessel’s fundamental quasi-symmetric function can be realized as the character of an irreducible crystal for the Lie superalgebra gln|n associated to its non-standard Borel subalgebra with a maximal number of odd isotropic simple roots. We als...

Generalized quasi-cyclic (GQC) codes with arbitrary lengths over the ring Fq + uFq, where u 2 = 0, q = p, n a positive integer and p a prime number, are investigated. By the Chinese Remainder Theorem, structural properties and the decomposition of GQC codes are given. For 1-generator GQC codes, minimal generating sets and lower bounds on the minimum distance are given. As a special class of GQC...

Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in X/S containing a given closed subsch...