ON SEMI-IFP RINGS

On Projective Modules over Semi-hereditary Rings

This theorem, already known for finitely generated projective modules[l, I, Proposition 6.1], has been recently proved for arbitrary projective modules over commutative semi-hereditary rings by I. Kaplansky [2], who raised the problem of extending it to the noncommutative case. We recall two results due to Kaplansky: Any projective module (over an arbitrary ring) is a direct sum of countably ge...

Potentially Semi-stable Deformation Rings

LetK/Qp be a finite extension and GK = Gal(K̄/K) the Galois group of an algebraic closure K̄. Let F be a finite field of characteristic p, and VF a finite dimensional F-vector space equipped with a continuous action of GK . The study of the deformation theory of Galois representations was initiated by Mazur [Ma], who showed that if VF has no non-trivial endomorphisms, then it admits a universal d...

Semi-simple Extensions of Rings

In this paper we investigate the conditions under which a given ring is a sub-ring of a semi-simple ring. For convenience, we say that a ring A is an extension of a ring B, if B is a sub-ring of A. It is found that the existence of a semi-simple extension is equivalent to the vanishing of the extension radical, a two-sided ideal defined analogously to the ordinary radical. In Theorem II we give...

Measures on Semi-Rings in R1 and Rk

On the Whitehead Determinant for Semi-local Rings

I referred her to [2], where Theorem 3.6 asserts that K1(R) is R /Ẽ, where Ẽ is the group generated by (1 + xy)/(1 + yx) with x, y in R and 1 + xy in R and where the last sentence in §3 says that Ẽ is not [R, R] in the case when R = M2(Z/2Z) is the ring of 2 by 2 matrices over a field of two elements (Z is the ring of integers). Moreover, in this case the group K1(R) is trivial while R/[R , R] ...