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Author(s):  
Wannarisuk Nongbsap ◽  
◽  
Dr. Madan Mohan Singh ◽  

In this paper, we present a public key scheme using Discrete Logarithm problem, proposed by Diffie and Hellman (DLP)[1], particularly known as the Computational Diffie-Hellman Problem (CDH)[12]. This paper uses the Elgamal encryption scheme [6] and extends it so that more than one message can be sent. The combination of Hill Cipher[14 ] and the property of the matrix ring 𝑴𝒏(𝒁𝒑), of being left m-injective over itself, where 𝒑 is a very large prime, are major contributions towards the proposal of this scheme.


Author(s):  
Roozbeh Hazrat ◽  
Lia Vaš

If [Formula: see text] is a directed graph and [Formula: see text] is a field, the Leavitt path algebra [Formula: see text] of [Formula: see text] over [Formula: see text] is naturally graded by the group of integers [Formula: see text] We formulate properties of the graph [Formula: see text] which are equivalent with [Formula: see text] being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of [Formula: see text] are also characterized in terms of the pre-ordered group properties of the Grothendieck [Formula: see text]-group of [Formula: see text]. If [Formula: see text] has finitely many vertices, we characterize when [Formula: see text] is strongly graded in terms of the properties of [Formula: see text] Our proof also provides an alternative to the known proof of the equivalence [Formula: see text] is strongly graded if and only if [Formula: see text] has no sinks for a finite graph [Formula: see text] We also show that, if unital, the algebra [Formula: see text] is strongly graded and graded unit-regular if and only if [Formula: see text] is a crossed product. In the process of showing the main result, we obtain conditions on a group [Formula: see text] and a [Formula: see text]-graded division ring [Formula: see text] equivalent with the requirements that a [Formula: see text]-graded matrix ring [Formula: see text] over [Formula: see text] is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group [Formula: see text] on the Grothendieck [Formula: see text]-group [Formula: see text]


Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2676
Author(s):  
Driss Bennis ◽  
Rachid El Maaouy ◽  
Juan Ramón García Rozas ◽  
Luis Oyonarte

Let A and B be rings, U a (B,A)-bimodule, and T=A0UB the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B. We show that when U is relative (weakly) compatible, we are able to describe the structure of GC-projective modules over T. As an application, we study when a morphism in T-Mod is a special GCP(T)-precover and when the class GCP(T) is a special precovering class. In addition, we study the relative global dimension of T. In some cases, we show that it can be computed from the relative global dimensions of A and B. We end the paper with a counterexample to a result that characterizes when a T-module has a finite projective dimension.


2021 ◽  
Vol 133 (1) ◽  
pp. 21-42
Author(s):  
Amir Kamal Amir ◽  
Nur Fadhilah ◽  
Ainun Mawaddah Abdal
Keyword(s):  

Author(s):  
M. Sivagami ◽  
T. Tamizh Chelvam

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be a positive integer and [Formula: see text] be the set of all [Formula: see text] matrices over [Formula: see text] For a matrix [Formula: see text] Tr[Formula: see text] is the trace of [Formula: see text] The trace graph of the matrix ring [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text][Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] The ideal-based trace graph of the matrix ring [Formula: see text] with respect to an ideal [Formula: see text] of [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] In this paper, we investigate some properties and structure of [Formula: see text] Further, it is proved that both [Formula: see text] and [Formula: see text] are Hamiltonian.


Author(s):  
Lixin Mao

Let [Formula: see text] be a formal triangular matrix ring, where [Formula: see text] and [Formula: see text] are rings and [Formula: see text] is a [Formula: see text]-bimodule. We give some computing formulas of homological dimensions of special [Formula: see text]-modules. As an application, we describe the structures of [Formula: see text]-tilting left [Formula: see text]-modules.


Author(s):  
S. T. Dougherty ◽  
Adrian Korban ◽  
Serap Şahinkaya ◽  
Deniz Ustun

AbstractIn 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 $$M_k(R)$$ M k ( R ) and the ring R,  where R is the commutative Frobenius ring. We show that codes over the ring $$M_k(R)$$ M k ( R ) are one sided ideals in the group matrix ring $$M_k(R)G$$ M k ( R ) G and the corresponding codes over the ring R are $$G^k$$ G k -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.


Author(s):  
WILLIAM WOODS

Abstract Let k be a finite field of characteristic p, and G a compact p-adic analytic group. Write kG for the completed group ring of G over k. In this paper, we describe the structure of the ring kG/P, where P is a minimal prime ideal of kG. We give an explicit isomorphism between kG/P and a matrix ring with coefficients in the ring ${(k'G')_\alpha }$ , where $k'/k$ is a finite field extension, $G'$ is a large subquotient of G with no finite normal subgroups, and (–) α is a “twisting” operation that preserves many desirable properties of the ring structure. We demonstrate the usefulness of this isomorphism by studying the correspondence induced between certain ideals of kG and those of ${(k'G')_\alpha }$ , and showing that this preserves many useful “group-theoretic” properties of ideals, in particular almost-faithfulness and control by a closed normal subgroup.


2021 ◽  
Vol 28 (01) ◽  
pp. 1-12
Author(s):  
Juan Huang ◽  
Hailan Jin ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Zhelin Piao

It is proved that for matrices [Formula: see text], [Formula: see text] in the [Formula: see text] by [Formula: see text] upper triangular matrix ring [Formula: see text] over a domain [Formula: see text], if [Formula: see text] is nonzero and central in [Formula: see text] then [Formula: see text]. The [Formula: see text] by [Formula: see text] full matrix rings over right Noetherian domains are also shown to have this property. In this article we treat a ring property that is a generalization of this result, and a ring with such a property is said to be weakly reversible-over-center. The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains. The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally. We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.


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