1-affine completeness of compatible modules

Peterson, Gary L.

doi:10.1007/s00012-016-0385-0

Cited by 4 publications

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“…If all factors of the composition series that are ring modules have order 2, then G and hence R are finite, since minimal R-modules that are nonring modules are always finite. If the series is a 1-affine complete chain and G i /G i+1 is a ring module coprime to G/G i , then [2,Theorem 3.3] gives us |G i /G i+1 | = 2…”

Section: Is a Composition Series Of R-ideals Of A Faithful Compatiblementioning

confidence: 99%

“…The statement of [2, Theorem 4.1] is not correct because, contrary to item (2) of this theorem, a composition series which is a 1-affine complete chain may have factors of order 2 that are not R-isomorphic. Nevertheless, such factors are indeed R-isomorphic if the nearring R is an automorphism nearring of G such as I(G) since the automorphisms generating R must act as the identity map on these factors.…”

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Erratum to “1-affine completeness of compatible modules”

Peterson

2017

Algebra Univers.

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Abstract. We correct a theorem in the original paper concerning when a composition series of a compatible nearring module is a 1-affine complete chain.The statement of [2, Theorem 4.1] is not correct because, contrary to item (2) of this theorem, a composition series which is a 1-affine complete chain may have factors of order 2 that are not R-isomorphic. Nevertheless, such factors are indeed R-isomorphic if the nearring R is an automorphism nearring of G such as I(G) since the automorphisms generating R must act as the identity map on these factors. To see, however, that this does not hold in general if the automorphism nearring assumption is dropped, let G be the additive group of Z 4 and R = C 0 (G). Then G > 2 > 0 is a 1-affine complete chain with G/ 2 and 2 coprime by [2, Theorem 3.5] since 2 is not contained in the sum of its maximal complements. A correct statement of the theorem is obtained by rewriting it in the following manner; the proof of this corrected result is obtained by making some modifications of the original one.

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Section: Is a Composition Series Of R-ideals Of A Faithful Compatiblementioning

confidence: 99%

mentioning

confidence: 99%