Periodic rings with commuting nilpotents

Autor: Hazar Abu-Khuzam, Adil Yaqub
Jazyk: angličtina
Rok vydání: 1984
Předmět:
Zdroj: International Journal of Mathematics and Mathematical Sciences, Vol 7, Iss 2, Pp 403-406 (1984)
Druh dokumentu: article
ISSN: 0161-1712
1687-0425
01611712
DOI: 10.1155/S0161171284000417
Popis: Let R be a ring (not necessarily with identity) and let N denote the set of nilpotent elements of R. Suppose that (i) N is commutative, (ii) for every x in R, there exists a positive integer k=k(x) and a polynomial f(λ)=fx(λ) with integer coefficients such that xk=xk+1f(x), (iii) the set In={x|xn=x} where n is a fixed integer, n>1, is an ideal in R. Then R is a subdirect sum of finite fields of at most n elements and a nil commutative ring. This theorem, generalizes the xn=x theorem of Jacobson, and (taking n=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume that In is a subring of R.
Databáze: Directory of Open Access Journals
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