Extensions of exchange rings
Given an ideal \(I\) of a ring \(R\), the determination of properties of \(R\) from those of \(I\) and \(R/I\) is known as an `extension problem'. Here the extension problem for exchange rings is solved: \(R\) is an exchange ring if and only if \(I\) and \(R/I\) are exchange rings and idempotents lift modulo \(I\). In the unital case, \(R\) is called an exchange ring provided the regular module \(R_R\) satisfies the (finite) exchange property of \textit{P. Crawley} and \textit{B. Jónsson} [Pac. J. Math. 14, 797-855 (1964; Zbl 0134.25504)] (cf. [\textit{R. B. Warfield}, jun., Math. Ann. 199, 31-36 (1972; Zbl 0238.16024)]). Non-unital exchange rings have not previously occurred, and so the author introduces them and develops their basic properties. With some further work, the extension theorem above is improved as follows: \(R\) is an exchange ring if and only if \(I\) and \(R/I\) are exchange rings and the canonical map \(K_0(R)\to K_0(R/I)\) is surjective. An interesting corollary characterizes semiperfect rings as exactly those semilocal rings \(R\) for which the canonical map \(K_0(R)\to K_0(R/J(R))\) is surjective. These results give completely algebraic generalizations of some extension results concerning C*-algebras of real rank zero, i.e., C*-algebras in which self-adjoint elements can be approximated arbitrarily closely by real linear combinations of orthogonal projections. The connection is a result of the author, \textit{K. C. O'Meara, E. Pardo}, and the reviewer, that a C*-algebra has real rank zero if and only if it is an exchange ring [Separative cancellation for projective modules over exchange rings, Isr. J. Math. (to appear)]. \textit{L. G. Brown} and \textit{G. K. Pedersen} proved [in J. Funct. Anal. 99, No. 1, 131-149 (1991; Zbl 0776.46026)] that if \(I\) is a closed ideal in a C*-algebra \(A\), then \(A\) has real rank zero if and only if \(I\) and \(A/I\) have real rank zero and projections lift modulo \(I\); if and only if \(I\) and \(A/I\) have real rank zero and the canonical map \(K_0(A)\to K_0(A/I)\) is surjective. These results are now obtained as corollaries of the author's extension theorems for exchange rings.
scientific article; zbMATH DE number 1104106
scientific article