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.

View Hutchins Errata Sheet (PDF)

"A module with a projective cover is a B-object (Bass module)" Over $\mathbb Z$, a countable direct sum of copies of $\mathbb Z$ is projective (hence has a projective cover) and yet it has quotients (namely $\mathbb Z_{p^\infty}$ with no maximal submodules. That is, the kernel of this projection is a proper submodule not contained in any maximal submodule.

Corrected in:

  • nobody. Custom. (2017) @ (obvious)

14.E is constructed for an arbitrary field, and 34.B says it is a G-ring. However, it may not be a G ring for fields of positive characteristics. The assertion is fine for a field of characteristic 0.

Corrected in:

  • Contributors. Database of Ring Theory. (2026) @ https://ringtheory.herokuapp.com/static/issue185.html

In Cozzens' paper on page 69 he states "Also note that $x_0$ and $x_0^2 \notin E$. This seems to be in error and results in the construction not being simple. See the corrected-in link for details.

Corrected in:

  • Contributors. Database of Ring Theory. (2026) @ https://ringtheory.herokuapp.com/static/issue188.html

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)

"ring is PCI ring on the right + ring is PCI ring on the left <==> ring is PCI ring on the right + ring is coherent on the left" According to Leroy, Jain & Lam, the <== direction is faulty. The ==> direction is fine.

Corrected in:

  • A. Leroy, S. Jain, and T. Lam. Ore extensions and V-domains. (2009) @ Remark 6.3

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.

Corrected in:

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

Example 16 is claimed to be local, but this is clearly not so. It has a quotient isomorphic to $\mathbb Z$ which is not local.

Corrected in:

  • nobody. Custom. (2017) @ (obvious)

According to the corrigendum, the right socle should be essential, not the left.

Corrected in:

  • W. K. Nicholson and M. F. Yousif. Corrigendum to the paper {A}nnihilators and the {CS} {C}ondition. (1998) @ main Theorem

The conclusion "not self-injective" requires additionally that the DVR is NOT $I$-adically complete. Therefore the sentence starting "A routine computation shows that..." does not hold. The linked citation is not a corrigendum for this issue, but it does witness the link with completeness.

Corrected in:

  • C. Faith. Self-injective rings. (1979) @ Theorem 4C p 161

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 McCoy condition. (2006) @ Section 3 p 138