On Modules for Which All Submodules Are Projection Invariant and the Lifting Condition

The notion of projection invariant subgroups was first introduced by Fuchs in [7]. We will define the module-theoretic version of the projection invariant subgroup. Let R be a ring and M a right R-module. We call a submodule N of M the projection invariant if every projection of M onto a direct summand maps N into itself, i.e. N is invariant under any projection of M. In this note, we give several characterizations to these class of modules that generalize the recent results in [14]. We also define and study the PI-lifting modules which is a generalization of FI-lifting module. It is shown that if each Mi is a PI-lifting module for all 1 ? i ? n, then M = ?n i=1Mi is a PI-lifting module. In particular, we focus on rings satisfying the following condition: (*) Every submodule of M is projection invariant. We prove that if R has the (*) property, then R ? R does not satisfy the (*) property

Rigid, quasi-rigid and matrix rings with (σ,0)-multiplication

Let R be a ring with an endomorphism σ. We introduce (σ, 0)-multiplication which is a generalization of the simple 0- multiplication. It is proved that for arbitrary positive integers m ≤ n and n ≥ 2, R[x; σ] is a reduced ring if and only if Sn,m(R) is a ring with (σ, 0)-multiplication

G -codes, self-dual G -codes and reversible G -codes over the ring B j, k

From Springer Nature via Jisc Publications RouterHistory: received 2020-09-25, accepted 2021-03-24, registration 2021-03-25, online 2021-05-03, pub-electronic 2021-05-03, pub-print 2021-09Publication status: PublishedAbstract: In this work, we study a new family of rings, Bj, k, whose base field is the finite field Fpr. We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study G-codes, self-dual G-codes, and reversible G-codes over this family. In particular, we show that the projection of a G-code over Bj, k to a code over Bl, m is also a G-code and the image under the Gray map of a self-dual G-code is also a self-dual G-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible G-code under the Gray map is also a reversible G2j+k-code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasi-G codes, which are the images of G-codes under the Gray map, are also Gs-codes for some s

Group matrix ring codes and constructions of self-dual codes

From Springer Nature via Jisc Publications RouterHistory: received 2021-01-30, rev-recd 2021-03-13, accepted 2021-03-19, registration 2021-03-20, pub-electronic 2021-04-02, online 2021-04-02, pub-print 2023-03Publication status: PublishedIn this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring Mk(R) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring Mk(R) are one sided ideals in the group matrix ring Mk(R)G and the corresponding codes over the ring R are Gk-codes of length kn. Additionally, we give a generator matrix for self-dual codes, which consist of the mentioned above matrix construction. We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes

Some results on δ-semiperfect rings and δ-supplemented modules

In [9], the author extends the definition of lifting and supplemented modules to ?-lifting and ?-supplemented by replacing "small submodule" with "?-small submodule" introduced by Zhou in [13]. The aim of this paper is to show new properties of ?-lifting and ?-supplemented modules. Especially, we show that any finite direct sum of ?-hollow modules is ?-supplemented. On the other hand, the notion of amply ?-supplemented modules is studied as a generalization of amply supplemented modules and several properties of these modules are given. We also prove that a module M is Artinian if and only if M is amply ?-supplemented and satisfies Descending Chain Condition (DCC) on ?-supplemented modules and on ?-small submodules. Finally, we obtain the following result: a ring R is right Artinian if and only if R is a ?-semiperfect ring which satisfies DCC on ?-small right ideals of R

Rigid, quasi-rigid and matrix rings with (σ, 0)-multiplication

Let R be a ring with an endomorphism ?. We introduce (?, 0)-multiplication which is a generalization of the simple 0- multiplication. It is proved that for arbitrary positive integers m ? n and n ? 2, R[x; ?] is a reduced ring if and only if Sn,m(R) is a ring with (?, 0)-multiplication