# 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.

## 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.

"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)

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.

Corrected in
• K. 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)

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