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Algebraic Structures I (Matrices, Groups, Rings, and Fields)
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
A principal ideal ring (or more specifically, a principal ideal domain (P.I.D.)) is a commutative ring A without zero divisors and with unit element 1 (that is, an integral domain) and in which every ideal is principal.
Signed ring families and signed posets
Published in Optimization Methods and Software, 2021
Kazutoshi Ando, Satoru Fujishige
For a signed poset and a vertex , when , define and when , define Also, if (or ), we define (or . We can easily see that and are ideals of and we call the positive principal ideal at v and the negative principal ideal at v of . (In fact, for a simple signed ring family on V and its corresponding signed poset on V we have and for .)
Some remarks on global analytic planar vector fields possessing an invariant analytic set
Published in Dynamical Systems, 2021
Let be an analytic vector field in having a coherent analytic invariant curve with F a Morse function such that the principal ideal in the ring is real. Then there are two analytic vector fields and in such that where F is a first integral of if and only if the function defined by is analytic. In particular the decomposition (9) holds provided is a smooth curve.
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