Ideal extensions of rings
type: Item
instance of: scholarly article
title: Ideal extensions of rings
zbMATH Open document ID: 0582.16011
DOI: 10.1007/BF01957021
published in: Acta Mathematica Hungarica
publication date: 1985-01-01T00:00:00Z
review text: A very instructive paper concerning a problem first solved by \textit{C. J. Everett} [Am. J. Math. 64, 363-370 (1942; Zbl 0060.076); cf. also \textit{L. Rédei}, Algebra, Vol. I (1967; Zbl 0191.005), {\S}{\S}52-54]. An ideal extension of a ring A by a ring B is considered as a triple (\(\phi\),R,\(\psi)\) such that \(0\to A\to^{\phi}R\to^{\psi}B\to 0\) is a short exact sequence. Two extensions (\(\phi\),R,\(\psi)\) and (\(\phi\) ',R',\(\psi\) ') of A by B are called equivalent if there exists an (equivalence) isomorphism \(\chi\) : \(R\to R'\) such that \(\phi\),\(\phi\) ',\(\chi\),\(\psi\) ',\(\psi\) form a commutative diagram. Then the author gives a new proof of Everett's theorem (and more) in three steps. (i) He constructs a special type of extensions of A by B by means of Everett sums, using in particular a function \(\theta\) of B into the translational hull \(\Omega\) (A) of A [cf. \textit{M. Petrich}, Semigroup Forum 1, 283-360 (1970; Zbl 0209.049)]. In this context the annihilator \({\mathcal A}(A)\) occurs for the first time. Then (ii) he shows that every extension of A by B is equivalent to an Everett sum and (iii) establishes a criterion for the equivalence of two Everett sums, which describes all equivalence isomorphisms. In the next section ({\S}5), some invariants of extensions as the character and the type, and in particular strict, pure and essential extensions are considered. The case \({\mathcal A}(A)=0\), which allows simpler constructions and stronger statements, is extensively investigated in {\S}6. Extensions of semiprime atomic rings are treated in {\S}7 in order to find conditions on such a ring A such that A admits only direct sum extensions by certain other rings B. Open problems on Everett sums of two rings and strict, pure and essential extensions are sketched at the end of the paper.
Mathematics Subject Classification ID: 16S20
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 3931199
zbMATH Keywords: ideal extension
zbMATH Keywords: short exact sequence
zbMATH Keywords: Everett's theorem
zbMATH Keywords: extensions
zbMATH Keywords: Everett sums
zbMATH Keywords: translational hull
zbMATH Keywords: annihilator
zbMATH Keywords: equivalence isomorphisms
zbMATH Keywords: invariants of extensions
zbMATH Keywords: strict, pure and essential extensions
zbMATH Keywords: semiprime atomic rings
zbMATH Keywords: direct sum
author: Mario Petrich
reviewed by: Hanns Joachim Weinert
MaRDI profile type: Publication
cites work: An Extension Theory for Rings
cites work: Extensions and obstructions for rings
cites work: The translational hull in semigroups and rings
cites work: Rings and semigroups. With an appendix by Richard Wiegandt
cites work: Q5829837
full work available at URL: https://doi.org/10.1007/bf01957021
OpenAlex ID: W2088239510
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description: scientific article; zbMATH DE number 3931199