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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
(Definition: A subset B of a ring A is a subring of A if B is a ring with respect to the binary operations + and · of A). Prove: Z is a subring of Q.Q is a subring of R.R is a subring of C.
Some Properties of Rough Pythagorean Fuzzy Sets
Published in Fuzzy Information and Engineering, 2021
Amal Kumar Adak, Davood Darvishi Salookolaei
A Pythagorean fuzzy set in a near-ring is said to be a Pythagorean fuzzy subring of if for all satisfies the following properties
A discrete variant of Farkas' lemma and related results
Published in Optimization, 2021
Assuming for all and for all , it is an exercise to show that for all , for any . Consequently, given the linear forms , for i = 1, …, m, and a point , it also holds and for all . It follows that the range of a linear form is an associative subring (even an ideal) of the ring R. Likewise, it holds for the linear mapping and for any that and for all .
Block conjugacy of irreducible toral automorphisms
Published in Dynamical Systems, 2019
Lennard F. Bakker, Pedro Martins Rodrigues
For two ideals I and J of , we define another ideal . The coefficient ring of I is defined to be , which is always an order of the field, i.e. a subring of that has rank n as a -module. The coefficient ring of I satisfies . Suppose that represents, as above, multiplication by β in I, with respect to a certain basis over ; then, if is given by a polynomial with rational coefficients, i.e. , then if and only if is an integer matrix. An obvious necessary condition for two ideals to be arithmetically equivalent is that they have the same coefficient ring. An ideal I with coefficient ring R is invertible if . An invertible ideal may always be generated, as an R-module, by no more than 2 elements of R [19].