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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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 inThe statement needed to be sharpened slightly, and the proof adjusted as described in the errata.
Corrected inMarot 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 inThe author claimed prime ideals of the ring of holomorphic functions on $\mathbb C$ are all maximal. This is false.
Corrected inElliger'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 inExample 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 inIt is claimed that semicommutative rings are McCoy, but this is false. The mistake was that semicommutativity does not pass to polynomial rings.
Corrected in