By E. R. Kolchin
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Extra resources for Differential Algebra and Algebraic Groups (Pure and Applied Mathematics 54)
By the choice of A, there exist x and y in R with a − axa ∈ I and a − aya ∈ J. Thus a − a(x + y − xay)a = (a − axa) − aya + (axa)ya and so a − a(x + y − xay)a ∈ I. Similarly a − a(x + y − xay)a ∈ J. Therefore, a − a(x + y − xay)a ∈ I ∩ J. But, since I ∩ J = 0, we have a − a(x + y − xay)a = 0. This contradicts the choice of A. Consequently, R is von Neumann regular. The converse is obvious. 22 If R is two-sided noetherian and each prime factor ring of R is von Neumann regular, then R/N (R) is von Neumann regular and N (R) is nilpotent.
Suppose B is not semisimple. There exists an F ⊂e B. We get an inﬁnite direct sum T1 = ⊕α∈Λ Fα with Fα ⊂e Bα . Then, every cyclic subfactor of R/T1 is proper, for otherwise, a complement of T in R will contain a copy of B, and that will give a contradiction. Hence R/T1 is of ﬁnite Goldie dimension. 7). Hence B is semisimple. 13 Let R be a right P CCS-ring.
From this, we conclude that S is a semiprimary ring. It follows that S is an artinian ring since S has left and right restricted minimum condition. Therefore R is a noetherian ring. A ring R is said to satisfy the restricted right socle condition (RRS for short) if for each essential right ideal E (= R), R/E has nonzero socle. A ring R is called a right QI-ring if each quasi-injective right R-module is injective. Boyle proved that a right QI-ring is right noetherian, and conjectured that every right QI-ring is hereditary .