Errata for ring theory literature

Below is a list of errors in the literature about rings. This list will probably not contain things as trivial as typos, but rather mistaken claims or serious gaps in proofs.

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Examples of Commutative Rings

Since we like Examples of Commutative rings so much, we plan to maintain a dedicated errata page for it. Thanks to the author H. C. Hutchins for providing the original errata.

The statement needed to be sharpened slightly, and the proof adjusted as described in the errata.

Corrected in
  • J. Lawrence. Erratum to:“A countable self-injective ring is quasi-Frobenius”(Proc. Amer. Math. Soc. 65 (1977), no. 2, 217--220). (1979) @ p 140

Marot asserted that a ring with the property "each ideal generated by a finite set of regular elements is principal" also has the property "each regular ideal is generated by a set of regular elements." Gilmer proposes a counterexample.

Corrected in
  • R. W. Gilmer. On prüfer rings. (1972) @ (entire note)

The author claimed prime ideals of the ring of holomorphic functions on $\mathbb C$ are all maximal. This is false.

Corrected in
  • M. Henriksen. On the prime ideals of the ring of entire functions.. (1953) @ Theorem 1(a)

Corrected in
  • W. J. Heinzer and others. Polynomial rings over a Hilbert ring. (1984) @ Whole article

Elliger's theorem was that a left-and-right self-injective simple ring must be Artinian, but this was shown to be false by Goodearl.

Example: Goodearl's simple self-injective von Neumann regular ring

Corrected in
  • K. Goodearl. Simple self-injective rings need not be artinian. (1974) @ main result

It is claimed that semicommutative rings are McCoy, but this is false. The mistake was that semicommutativity does not pass to polynomial rings.

Corrected in
  • P. P. Nielsen. Semi-commutativity and the {M}c{C}oy condition. (2006) @ Section 3 p 138