Journal article
On infinite MacWilliams rings and minimal injectivity conditions
Proceedings of the American Mathematical Society, Vol.150(11), p.4575
11/01/2022
DOI: 10.1090/proc/15929
Abstract
We provide a complete answer to the problem of characterizing left Artinian rings which satisfy the (left or right) MacWilliams extension theorem for linear codes, generalizing results of Iovanov [J. Pure Appl. Algebra 220 (2016), pp. 560–576] and Schneider and Zumbrägel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961] and answering the question of Schneider and Zumbragel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961]. We show that they are quasi-Frobenius rings, and are precisely the rings which are a product of a finite Frobenius ring and an infinite quasi-Frobenius ring with no non-trivial finite modules (quotients). For this, we give a more general “minimal test for injectivity” for a left Artinian ring A: we show that if every injective morphism \Sigma _k\rightarrow A from the k’th socle of A extends to a morphism A\rightarrow A, then the ring is quasi-Frobenius. We also give a general result under which if injective maps N\rightarrow M from submodules N of a module M extend to endomorphisms of M (pseudo-injectivity), then all such morphisms N\rightarrow M extend (quasi-injectivity) and obtain further applications.
Details
- Title: Subtitle
- On infinite MacWilliams rings and minimal injectivity conditions
- Creators
- Miodrag Cristian Iovanov
- Resource Type
- Journal article
- Publication Details
- Proceedings of the American Mathematical Society, Vol.150(11), p.4575
- DOI
- 10.1090/proc/15929
- NLM abbreviation
- Proc Natl Acad Sci U S A
- ISSN
- 0002-9939
- eISSN
- 1088-6826
- Publisher
- American Mathematical Society
- Number of pages
- 12
- Grant note
- DOI: 10.13039/100000893, name: Simons Foundation, award: 637866
- Language
- English
- Date published
- 11/01/2022
- Academic Unit
- Mathematics
- Record Identifier
- 9984302601702771
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