On regular rings with involution
This paper is concerned with *-regular rings, i.e., (von Neumann) regular rings with involution in which every principal right ideal is generated by a projection. Handelman has asked whether every *-regular ring is directly finite (that is, \(xy=1\) implies \(yx=1)\). Here the authors prove that every *-regular ring is at least *-finite, that is, \(xx^*=1\) implies \(x^*x=1\). In a related result, they prove that any ring of positive characteristic with a proper involution (that is, \(xx^*=0\) implies \(x=0)\) must be *-finite. The remaining major result of the paper concerns the regular ring of a finite Rickart \(C^*\)-algebra (i.e., a *- finite \(C^*\)-algebra in which the right annihilator of any element is a principal right ideal generated by a projection). \textit{D. Handelman} proved that any finite Rickart \(C^*\)-algebra T is a *-subring of an \(\aleph_ 0\)-continuous *-regular ring Q such that the projections in Q all lie in T [Studies in Analysis, Adv. Math., Suppl. Stud., Vol. 4, 171- 196 (1979; Zbl 0511.46054); Theorem 2.1]. He later showed that if the 2\(\times 2\) matrix algebra \(M_ 2(T)\) is a finite Rickart \(C^*\)- algebra, then T is a classical order in Q, that is, all non-zero-divisors in T are invertible in Q and all elements of Q can be expressed in the forms \(ab^{-1}\) and \(c^{-1}d\) for \(a,b,c,d\in T\) with b and c non- zero-divisors [Adv. Math. 48, 1-15 (1983; Zbl 0524.46037); Theorem 8.3]. This result was used by \textit{D. Handelman}, \textit{J. Lawrence}, and the reviewer to show that if T has no one-dimensional representations, then T is a classical order in Q [Mem. Am. Math. Soc. 234 (1980; Zbl 0435.16005), Theorem III.16.9]. (A simplification of these arguments, due to Berberian, appears in another paper of \textit{D. Handelman} [Rocky Mt. J. Math. 11, 337-381 (1981; 473.16013), Corollary 1.2].) In the present paper, the authors give a very nice short direct proof that T is always a classical order in Q. This is done by showing that T has a classical ring of quotients Q' which is *-regular, and then observing that Q' may be identified with Q.
von Neumann regular rings with involution
scientific article; zbMATH DE number 3847594
scientific article