Vol. 58, No. 2, 1975

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Three questions on duo rings

Hans-Heinrich Brungs

Vol. 58 (1975), No. 2, 345–349

A ring R (with unit element) is called a duo ring if every one-sided ideal is two-sided. This is equivalent with the existence of elements rand r′′ in R with rs = sr,sr = r′′s for elements r,s in R. We will discuss in this note the following three problems: (A) Is the localization at a prime ideal P of a duo ring again a duo ring? (B) Is in a duo ring the P-component of zero equal to the right (left) P-component of zero? (C) Is in a noetherian duo domain the semi group of ideals (under multiplication) commutative? The answer to all three questions is “no” in general, but “yes” for (A) and (B) in the noetherian case, and “yes” for (C) if R is integrally closed in its division ring of quotients.

Mathematical Subject Classification
Primary: 16A66
Published: 1 June 1975
Hans-Heinrich Brungs