Some noncommutative rings constructed on the base of polynomials x² + tx + 1 and their zero divisors
Journal Title: Bulletin de la Société des sciences et des lettres de Łódź, Série: Recherches sur les déformations - Year 2017, Vol 0, Issue 1
Abstract
For any Galois field F = GF(pn) we construct some ring extension R(F) of order p4n. Such construction may be applied also for any infinite field F with char F 6= 0. Then, for any element of R(F) we give necessary and sufficient condition to be a zero divisor. With additional assumptions, we make some variations on this condition. In special cases we are able to calculate easily the number of all zero divisors and of idempotents and nilpotents of degrre 2. The method used to construct R(F) is the following. First we find t 2 F, such that the polynomial x2 + tx + 1 does not have roots in F (such t-s exist!). Then we take the 4-dimensional F-vector space with basic elements 1, i, j, k, where i, j, k not belonging to F are roots of x2 +tx+1 in the ring extension, and ji = k t. Thus multiplication of i, j, k (and hence in all ring) is some generalization of multiplication in the real Hamilton quaternions. In consequence, we have got wide class of noncommutative rings. It is known very much on noncommutative rings of smaller order, e.g. in 1994 J. B. Derr, G. F. Orr, and P. S. Peck classified all noncommutative rings of order p4, using radical as a helpful tool. Thus, in the particular case n = 1, each of the rings constructed here must be of one kind given by Derr and the others. Our consideration is more general. It turns out that selected properties of R(F) depend on char F and on t2 22 is a square in F or not. To get these and other results, we use some properties of multiplicative subgroup of nonzero squares in GF(pn) and of the polynomial x2+tx+1. All contents is provided with examples illustrating general situation or special cases.
Authors and Affiliations
Jan Jakóbowski
Keywords
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- EP ID EP236248
- DOI 10.26485/0459-6854/2017/67.1/4
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