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Ring Theory
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
which has a similar structure to that of ℤ[x]/〈xn−1〉. This observation provides an alternate construction of the former ring, which can be seen through the concept of a ring homomorphism. A ring homomorphism is a map between rings that respects both the additive structure and multiplicative structure of these rings.
Elements of topology and homology
Published in Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama, Computational Topology for Biomedical Image and Data Analysis, 2019
Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama
A homomorphism is a structure-preserving function in the domains of algebra, discrete mathematics, groups, rings, graphs, and lattices. Structure-preserving functions between groups preserve their natural (algebraic) structure.
Field Extensions
Published in Richard A. Mollin, Algebraic Number Theory Second, 2011
where the first quotient is that of a ring modulo an ideal, and the second quotient is as a free abelian group modulo a cyclic subgroup. We may also calculate DF/P by counting its elements. Although there are other means of doing this, we explore this avenue for its instructive and illustrative value. First, we observe that P is maximal, for if u+v10∉P={2a+b10:a,b∈Z},
General remainder theorem and factor theorem for polynomials over non-commutative coefficient rings
Published in International Journal of Mathematical Education in Science and Technology, 2020
A. Cuida, F. Laudano, E. Martinez-Moro
In this final section we will point to some known results for evaluating and dividing on skew polynomial rings. This will show the potential of the topic all around the mathematical instruction, note that also this approach has been used actually in applied mathematics as for example in coding theory (see for example Martínez-Penas, 2018). We will rectrict ourselves to skew polynomial rings over fields with no derivation but the same results can be stated easily for general skew polynomial rings over division rings (Lam & Leroy, 1988). Let F be a field and σ an automorphism of F. The skew polynomial ring is the set of polynomials over the field F where the addition is defined as the usual one in the polynomial ring and the multiplication is defined as follows Indeed it a non-commutative ring unless σ is the identity. One can see that the Euclidean algorithm holds for right division which implies that R is a left principal ideal domain (Lam & Leroy, 1988). So following the previous sections we can evaluate a (left) polynomial on . Note that the obviuous ‘evaluation’ is wrong since it does not take into account the action of σ, but if we define whe have the following Factor Theorem
Least-squares solutions of the reduced biquaternion matrix equation AX=B and their applications in colour image restoration
Published in Journal of Modern Optics, 2019
Definition RB matrix: The RB matrix has four components (3), and it is often represented by the linear combination of two complex matrices using – forms as follows: where are complex matrices and and are the real matrices. The set of matrices with RB entries, which is denoted by , with usual matrix addition and multiplication is a ring with unity. There exist three kinds of conjugates of and they are and A matrix is a transpose of . A matrix is Moore–Penrose generalized inverse of . Also, is called conjugate transpose according to the th conjugate of . Some algebraic operations of the RB matrices are listed as follows (3, 6):
Finite-time estimation for linear time-delay systems via homogeneous method
Published in International Journal of Control, 2019
In order to simplify the analysis, let us introduce the delay operator δ (see Williams & Zakian, 1977) such that x(t − kh) = δkx(t), k ≥ 0. Let be the polynomial ring of δ over the field . After having introduced the delay operator δ, system (1) can be written as follows: where , and .