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Extra info for Algebra, K-Theory, Groups, and Education: On the Occasion of Hyman Bass's 65th Birthday

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The module _FrR(X) is -called the- standard free right R-module on-X. _dx EX Xrz (in) with_ri(m) E R, almost all rx (rn) being 0, and that nt-= n in-FrR(X) ifand only r(m) = r(n) for all x in X. In particular, if X =-{x} has one member, FrR({x}) = xR is a cyclic right module isomorphic to R. ) Properly speaking, the set X is not actually a subset of FrR(X). However, there is a canonical embedding tx : X FrR(X), the element tx(x) being defined by the requirements that r ii if y = x, )) v ‘tx `s" - 10 if y- s.

Let QU(R) be the set of quasi-invertible elements in R. Show that QU(R) is a group under t. Show that if R is actually a ring, then there is a group isomorphism ER : QU(R) U(R) given by eR(r) =1+ r. Verify that U(Ti)/ETz (tQU(R)) has order 2 for any nonunital ring R. 12) we proved that a commutative domain 0 can be embedded in a field of fractions. One might hope that the method could be extended to show that if R is a noncommutative domain, then R can be embedded in a division ring of fractions.

Show also that if M is a right R[T]-module, then M is a right R-module and the map a given by am = mT is an R-module endomorphista of M. Informally, T is said to act as a. Suppose that M and N are right R[7]-modules, with T acting as a and )3 respectively. Prove that 7r :M—Nis an R[7]-module (Note. 7 This exercise reviews some facts from elementary linear algebra concerning the matrix representation of a linear map and the way in which it is affected by a change of basis. 7)) for arbitrary rings, but it will be useful to invoke them for illustrative purposes beforehand.

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