1 Correction to: Boll Unione Mat Ital https://doi.org/10.1007/s40574-018-0161-5
The paper [3] by the authors contains an error. In Theorem 14, the last line of the proof of Claim B is incorrect when \(k=\frac{n}{2}\) and \(u=1\). If \(|D/pD| \ne 2\), the proof can be easily fixed by decomposing \(1=v+w\) with v, \(w \in D {\setminus } pD\). In the case where \(k = \frac{n}{2}\) and \(|D/pD|=2\) this is not possible, and the statement characterizing when \(2 \in \mathsf {L} (\alpha )\) needs to be changed.
The correct characterization for \(2 \in \mathsf {L} (\alpha )\) in Theorem 14 should read:
If\(k \ge 1\), then\(2 \in \mathsf {L} (\alpha )\)if and only if one of the following non-exclusive statements holds:
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\(k < \frac{n}{2}\),
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nis even and\(|D/pD| \ne 2\),
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nis even and either\(n=2\)or\(k \ne \frac{n}{2}\).
The characterization of \(2 \in \mathsf {L} (a)\) is used in description of other lengths in the proof of Theorem 14. This statement also has to be corrected if \(|D/pD|=2\); we refer to [1, Theorem 2.18]. In Corollary 16 the assumption \(|D/pD| \ne 2\) for all primes p of D has to be added.
The mistake was first noticed by A. McQueen, see [2], and later corrected by M. Axtell, N. R. Baeth, and J. Stickles. We refer to their paper [1], in particular Lemmas 2.6 and 2.7, for proofs in the case where \(|D/pD|=2\).
We sincerely regret the error and thank M. Axtell, N. R. Baeth, and J. Stickles for pointing it out to us.
Change history
20 March 2018
The paper [3] by the authors contains an error.
References
Axtell, M., Baeth, N.R., Stickles, J.: Factorizations in self-idealizations of PIRs and UFRs. J. Boll. Unione Mat. Ital. (2016)
Bachman, D., Baeth, N.R., McQueen, A.: Factorizations of upper triangular Toeplitz matrices. Boll. Unione Mat. Ital. 8(2), 131–150 (2015)
Chang, G.W., Smertnig, D.: Factorization in the self-idealization of a PID. Boll. Unione Mat. Ital. (9) 6(2), 363–377 (2013)
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Chang, G.W., Smertnig, D. Correction to: Factorization in the self-idealization of a PID. Boll Unione Mat Ital 12, 515–516 (2019). https://doi.org/10.1007/s40574-018-0161-5
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DOI: https://doi.org/10.1007/s40574-018-0161-5