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• W2013824385 abstract "Let R be a commutative ring. In (1), it was proved that a ring R with noetherian total quotient ring is self-injective if and only if the endomorphism ring of every ideal is commutative. We prove here that if the ring is coherent and is its own total quotient ring, then R is self-injective if and only if Hom(I, I) = R for every ideal I of R. In (1), we discussed the commutativity of endomorphism rings of ideals. There, we proved that a ring with noetherian total quotient ring is self-injective if and only if the endomorphism ring of every ideal is commutative. The generalization of this to nonnoetherian total quotient rings seems extremely difficult. We will call an element x of R a stable element if every nonzero homomorphism (x) to R is a multiplication by an element of R. We remark that if every element of R is stable, then Hom (I, I) is commutative for every ideal I of R. The converse is in general not true. But, we have some partial answers in this direction. If the ring is coherent and is its own total quotient ring and if Hom (I, I) = R for every ideal generated by 2 elements then every element of R is stable. On the other hand we have an example of a noncoherent ring which is its own total quotient ring which has a nonstable element but Hom(I, I) is commutative for every ideal I of R. We start with an example of a ring which is not self-injective, but Hom(I, I) is commutative for every ideal I of R. EXAMPLE. R = K[x , x2,... ]/(X2, x2, .. . ) where xl, x2, ... are indeterminates. R is the inductive limit of the rings K[x,x2,... , xn]/(xI x) n = 1, 2, .. ., and each of them is self-injective. It follows that Hom(I,I) is commutative for every ideal I. It is easy to see that R is not self-injective. We note that in this example R has the following properties. Every element of R is stable, Hom (I, I) = R for every finitely generated ideal I. R is further quasi-local and the zero ideal of R is irreducible. We will presently see that these are not isolated phenomena. We note first the following general result. PROPOSITION 1. If every element of R is stable, then Hom(I, I) is commutative for every ideal I of R. Received by the editors May 30, 1975. A MS (MOS) subject classifications (1970). Primary 13C05; Secondary 13A99." @default.
• W2013824385 created "2016-06-24" @default.
• W2013824385 creator A5048013771 @default.
• W2013824385 date "1976-02-01" @default.
• W2013824385 modified "2023-09-26" @default.
• W2013824385 title "Commutativity of endomorphism rings of ideals. II" @default.
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• W2013824385 doi "https://doi.org/10.1090/s0002-9939-1976-0401731-6" @default.
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