Algebra in a Local Topos With Applns to Ring Theory by F. Borceux, et al.,

By F. Borceux, et al.,

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N 0 = xK = (p"x 2 , x 3 - pmx•, x'- p•mx•, ... ). 1 8 is capable. 2 are of the form N = (pkx 1, N 0 ), n - m :s; k :s; n. 3 PfN :_. P/N 0 • But P/N :;;;; R = (b I b 2 = pmb, char b = pm+r•). 9 Calculation of the families of rings R far which Rfm-(R) is the null ring Zll of order 4 with an elementary abelian additive group. 4 Zll is capable. 8 each family of rings R with Rfm-(R) :;;;; Zll contains a 2-algebra, and hence without loss of generality we may perform all computations over GF(2). Let P be the free 2-algebra on indeterminates {x, y}, K = N 1 = P•, so N 0 = P 3 • Thus N 1 fN 0 has a basis consisting of all monomials oflength 2 from P.

This choice of a,, m < i :::;; n, completes a basis for S, for the set has the proper cardinality and spans 8. Extend cp by linearity to all of R. The consistency of the family isomorphisms implies cp preserves products and hence is an isomorphism of R onto S. 7 CoROLLARY Let RandS be algebras over a field Fin the same family of algebras over F, with dim R < dimS < oo. Then 8 ~ R Ef) N, where N is a null algebra over F. Proof Let N be the null algebra of dimension dim(8) - dim(R) over F. 4, Rt.

Suppose that xz is right quasi-regular, so that there exists b E R such that xz b xzb = 0. Then R· Thus RN s; R· Now suppose R satisfies the descending chain condition. Then N is nilpotent, and R s; N, so R is nilpotent. 5 R 2 s; R· If, on the other hand, R 2 s; R, then R/<1> R, along with RfN, is null. But RfN is semi-simple, so R = N is nilpotent.

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