Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes
Let S be a unital ring, S[t; σ, δ] a skew polynomial ring where σ is an injective endomorphism and δ a left σ -derivation, and suppose f ε S[t; σ, δ] has degree m and an invertible leading coefficient. Using right division by f to define the multiplication, we obtain unital nonassociative algebras...
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| Format: | Article |
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American Institute of Mathematical Sciences
2017
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| Online Access: | https://eprints.nottingham.ac.uk/38812/ |
| _version_ | 1848795696692461568 |
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| author | Pumpluen, Susanne |
| author_facet | Pumpluen, Susanne |
| author_sort | Pumpluen, Susanne |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | Let S be a unital ring, S[t; σ, δ] a skew polynomial ring where σ is an injective endomorphism and δ a left σ -derivation, and suppose f ε S[t; σ, δ] has degree m and an invertible leading coefficient. Using right division by f to define the multiplication, we obtain unital nonassociative algebras Sf on the set of skew polynomials in S[t; σ, δ] of degree less than m. We study the structure of these algebras. When S is a Galois ring and f base irreducible, these algebras yield families of finite unital nonassociative rings A, whose set of (left or right) zero divisors has the form pA for some prime p. For reducible f, the Sf can be employed both to design linear (f, σ, δ)-codes over unital rings and to study their behaviour. |
| first_indexed | 2025-11-14T19:36:12Z |
| format | Article |
| id | nottingham-38812 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:36:12Z |
| publishDate | 2017 |
| publisher | American Institute of Mathematical Sciences |
| recordtype | eprints |
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| spelling | nottingham-388122020-05-04T18:58:40Z https://eprints.nottingham.ac.uk/38812/ Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes Pumpluen, Susanne Let S be a unital ring, S[t; σ, δ] a skew polynomial ring where σ is an injective endomorphism and δ a left σ -derivation, and suppose f ε S[t; σ, δ] has degree m and an invertible leading coefficient. Using right division by f to define the multiplication, we obtain unital nonassociative algebras Sf on the set of skew polynomials in S[t; σ, δ] of degree less than m. We study the structure of these algebras. When S is a Galois ring and f base irreducible, these algebras yield families of finite unital nonassociative rings A, whose set of (left or right) zero divisors has the form pA for some prime p. For reducible f, the Sf can be employed both to design linear (f, σ, δ)-codes over unital rings and to study their behaviour. American Institute of Mathematical Sciences 2017-08-01 Article PeerReviewed Pumpluen, Susanne (2017) Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes. Advances in Mathematics of Communications, 11 (3). pp. 615-634. ISSN 1930-5338 Skew Polynomial Ring Ore Polynomials Nonassociative Algebra Commutative Finite Chain Ring Generalized Galois Rings Linear Codes (f σ δ)-codes Skew-constacyclic Codes http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=14505 doi:10.3934/amc.2017046 doi:10.3934/amc.2017046 |
| spellingShingle | Skew Polynomial Ring Ore Polynomials Nonassociative Algebra Commutative Finite Chain Ring Generalized Galois Rings Linear Codes (f σ δ)-codes Skew-constacyclic Codes Pumpluen, Susanne Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes |
| title | Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes |
| title_full | Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes |
| title_fullStr | Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes |
| title_full_unstemmed | Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes |
| title_short | Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes |
| title_sort | finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes |
| topic | Skew Polynomial Ring Ore Polynomials Nonassociative Algebra Commutative Finite Chain Ring Generalized Galois Rings Linear Codes (f σ δ)-codes Skew-constacyclic Codes |
| url | https://eprints.nottingham.ac.uk/38812/ https://eprints.nottingham.ac.uk/38812/ https://eprints.nottingham.ac.uk/38812/ |