Ideal extensions of rings
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.
scientific article; zbMATH DE number 3931199
scientific article