Ring (mathematics)

About: Ring (mathematics) is a research topic. Over the lifetime, 19980 publications have been published within this topic receiving 233849 citations. The topic is also known as: ring possibly without identity.

Key Agreement Protocols Based on Multivariate Algebraic Equations on Quaternion Ring.

Abstract: In this paper we propose new key agreement protocols based on multivariate algebraic equations. We choose the multivariate function F(X) of high degree on non-commutative quaternion ring H over finite field Fq. Common keys are generated by using the public-key F(X). Our system is immune from the Grobner bases attacks because obtaining parameters of F(X) to be secret keys arrives at solving the multivariate algebraic equations that is one of NP complete problems .Our protocols are also thought to be immune from the differential attacks and the rank attacks.

On Nilpotent Elements of Skew Polynomial Rings

Abstract: We study the structure of the set of nilpotent elements in skew polynomial ring R [ x ; ], when R is an -Armendariz ring. We prove that if R is a nil -Armendariz ring and t = I R , then the set of nilpotent elements of R is an -compatible subrng of R . Also, it is shown that if R is an -Armendariz ring and t = I R , then R is nil -Armendariz. We give some examples of non -Armendariz rings which are nil -Armendariz. Moreover, we show that if t = I R for some positive integer t and R is a nil -Armendariz ring and nil ( R [ x ][ y ; ]) = nil ( R [ x ])[ y ], then R [ x ] is nil -Armendariz. Some results of [3] followas consequences of our results.

A nonidentity for right alternative rings

Abstract: In rings of characteristic not two, (1) implies (2) [2], while in rings of characteristic two this is not the case [3]. In the following note we establish that in the free strongly right alternative ring R generated by a and b we have ((a, b), a, b) O0. We also know from previous work [2, Lemma 5(i)] that ((a, b), a, b)2 = 0 in R. While it was already known from theoretical considerations that the free right alternative ring of characteristic not two on two or more generators would have to have at least one element x HZ 0 such that x2 = 0, since otherwise the Main Theorem of [2 ] implies it would have to be alternative, which we know is not the case, this enables one to locate such an element specifically for the first time. In order to prove that ((a, b), a, b)0 in R, it suffices to give an example of a strongly right alternative algebra in which this is not an identity. Our example has basis elements ql, , qg. The only nonzero products of basis elements are the following twelve: q2= q3, qlq2 2 =q5, q2ql = q6, q2=q4, q2q7 = q9, q3q2 =-q7, q4ql =-q8, q4q3 =-q9, q5ql = q7, q6q2==qs, q6q5= q9, and q8ql=q9. From this information one may construct the multiplication table for the basis elements of the algebra. To see that the algebra A thus defined is right alternative, it is sufficient to verify (1) whenever x, y, and z are basis elements. The identity must then hold for all elements because of the linearity of the associator. It can easily be verified that q7, q8, and qg are in the nucleus and so (1) is automatically satisfied if any of these appear. It can also be verified that qiR R = 0 = qjR2 for i = 3, 5. Hence (1) holds if x = q3 or

Articulated ring puzzle

Abstract: A multi-component puzzle comprised of ring-forming components which are rotatable about selected orthogonally related axes to move individual components into various relationships. The components are co-acting and constrained to move about each other without the need for a central framework. The absence of a central framework permits a variety of puzzle forms, including intersecting rings, lattices, cubes and combinations of these forms.

On isomorphisms between centers of integral group rings of finite groups

Abstract: '. For finite nilpotent groups G and G', and a G-adapted ring S (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings SG and SG' is monomial, i.e., maps class sums in SG to class sums in SG' up to multiplication with roots of unity. As a consequence, G and G' have identical character tables if and only if the centers of their integral group rings ZG and ZG' are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.