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\(E\)-rings and related structures. (English) Zbl 0988.16021

Chapman, Scott T. (ed.) et al., Non-Noetherian commutative ring theory. Dordrecht: Kluwer Academic Publishers. Math. Appl., Dordr. 520, 387-402 (2000).
If \(R\) is a unital ring, the left multiplications in \(R\) form a subring of the ring of additive endomorphisms of \(R\). If this subring is the full ring of additive endomorphisms, then \(R\) is called an \(E\)-ring. Similarly, an \(R\)-module \(M\) is called an \(E\)-module if the group of \(R\)-morphisms from \(R\) to \(M\) is the full group of additive homomorphisms from \(R\) to \(M\).
This paper is a survey without proofs of the major properties of \(E\)-rings and \(E\)-modules and their applications. There is also a useful list of unsolved problems in the area.
In the final section, the author presents some new results: \(R\) is a two-sided \(E\)-ring if the ring of additive endomorphisms is generated by left and right multiplications by elements of \(R\). The finite rank two-sided \(E\)-rings are classified.
For the entire collection see [Zbl 0964.00012].

MSC:

16S50 Endomorphism rings; matrix rings
16W20 Automorphisms and endomorphisms
20K15 Torsion-free groups, finite rank
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
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