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Elementary Algebra
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
If R is a commutative ring with unity, thenevery maximal ideal is a prime ideal.R is a field if and only if the only ideals of R are R and {0}.I is a maximal ideal if and only if R/I is a field.
On the finiteness of accessibility test for nonlinear discrete-time systems
Published in International Journal of Control, 2021
Mohammad Amin Sarafrazi, Ewa Pawłuszewicz, Zbigniew Bartosiewicz, Ülle Kotta
Let be the commutative ring of polynomials in with coefficients in the ring . For a fixed k>n, let be the number of all submatrices of . Denote by the numerator of determinants of these submatrices as polynomials in . So, each can be written as where each is a monomial in with coefficient in . The index in (12) denotes the number of distinct monomials in .
MVW-rigs and product MV-algebras
Published in Journal of Applied Non-Classical Logics, 2018
Alejandro Estrada, Yuri A. Poveda
Given the MV-algebra , through the Mundici functor (Mundici, 1986) we get the -group which is isomorphic to the set of continuous functions of in that have the property that each of them is constituted by finite linear polynomials with integer coefficients and that is contained in the -ring . Thus we can take the -ring generated by in which we will call . This -ring is isomorphic to -ring of functions in each consisting of finite polynomials of . is a -commutative ring. Given a strong unit u of such that , we take . This MV-algebra with the usual product of functions is a commutative MVW-rig denoted by . Note that (Zuluga, 2017).
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 abstract algebra, a division ring (cf. Lam & Leroy, 1988; Martínez-Penas, 2018), also called a skew field (cf. Smits, 1968), is a ring in which division is possible. Specifically, it is a non-zero ring in which every non-zero element a has a multiplicative inverse, i.e. an element x with ax = xa = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all non-zero elements. A division ring is a type of non-commutative ring.